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Linear Algebra and its Applications 458 (2014) 429–453
Contents lists available at ScienceDirect
Linear Algebra and its Applications
www.elsevier.com/locate/laa
Ordering trees and graphs with few cycles by
algebraic connectivity
Nair Abreu a, Claudia Marcela Justel b, Oscar Rojo c, Vilmar Trevisan d,∗
a Universidade Federal do Rio de Janeiro, Brazilb Instituto Militar de Engenharia, Rio de Janeiro, Brazilc Department of Mathematics, Universidad Católica del Norte, Antofagasta, Chiled Instituto de Matemática, UFRGS, Porto Alegre, Brazil
a r t i c l e i n f o a b s t r a c t
Article history:Received 8 January 2013Accepted 9 June 2014Available online 28 June 2014Submitted by S. Kirkland
MSC:05C50
Keywords:Algebraic connectivityGraph orderingLaplacian matrixTree ordering
Several approaches for ordering graphs by spectral parameters are presented in the literature. We can find graph orderings either by the greatest eigenvalue (spectral radius or index) or by the sum of the absolute values of the eigenvalues (the energy of a graph) or by the second smallest eigenvalue of the Laplacian matrix (the algebraic connectivity), among others. By considering the fact that the algebraic connectivity is related to the connectivity and shape of the graphs, several structural properties of graphs relative to this parameter have been studied. Hence, a large number of papers about ordering graphs by algebraic connectivity, mainly about trees and graphs with few cycles, have been published. This paper surveys the significant results concerning these topics, trying to focus on possible points to be investigated in order to understand the difficulties to obtain partial orderings via algebraic connectivity.
© 2014 Elsevier Inc. All rights reserved.
* Corresponding author.E-mail addresses: [email protected] (N. Abreu), [email protected] (C.M. Justel),
[email protected] (O. Rojo), [email protected] (V. Trevisan).
http://dx.doi.org/10.1016/j.laa.2014.06.0160024-3795/© 2014 Elsevier Inc. All rights reserved.
430 N. Abreu et al. / Linear Algebra and its Applications 458 (2014) 429–453
1. Introduction
Ordering objects is a natural question and this is a particularly important problem in combinatorics. Accordingly, ordering of graphs is a particular class of problems that is extensively studied. Two obvious parameters, number of vertices and edges, are not enough to order graphs as there are too many objects with the same classification. What is usually understood by ordering graphs is to give a (total) order of all graphs with the same number of vertices, according to some invariant.
Most papers studying graph ordering use a spectral parameter to compare elements. For ordering results up to 1988, we refer to Cvetkovic̀ et al. [7]. From that date, other results about the subject may be highlighted. For example, Hofmeister [29], Chang and Huang [6], Lin and Guo [41] and Li et al. [39] order trees by the largest eigenvalues of the adjacency matrices in a special class of trees. Zhang and Chen [71], Yu, Lu and Tang [69], Belardo, Marzi and Simic̀ [3], Guo [21] and Li et al. [40] order graphs by adjacency and Laplacian spectral indices. Rada and Uzcàtegui [50] give an ordering of chemical trees by Randic̀ index. Ilic̀, in [30], Zhang et al. [74] and He and Le [27] investigate an ordering of trees by Laplacian coefficients. Xu [68] determines extremal trees relative to Harary indices. Wang [64] presents an order of Huckel trees according to minimal energies. Wang and Kang [65] and Guo [20] study ordering of graphs by energy and Laplacian energy. The Laplacian energy was also chosen for ordering trees by Trevisan et al. [63] and Fritscher et al. [18,17].
Among all papers dealing with this issue, several of them refer to the subject of ordering or finding extremal trees by algebraic connectivity. The first paper on this particular topic is due to Grone and Merris [25] and was published in 1990. After that, a great number of papers were published and, only in recent years, we can find around fifty references about the ordering of graphs via algebraic connectivity. For example, Zhang and Liu [73] order trees with nearly perfect matchings by algebraic connectivity. Biyikoglu and Leydold [4] investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. Shao et al. [58] order trees and connected graphs by algebraic connectivity and Li et al. [36] extend their results. Lal et al. [35] also study this invariant on unicyclic graphs and connected graphs with certain number of pendant vertices. For a given n, Wang et al. [66] determine the graph with the largest algebraic connectivity among graphs with diameter at most 4. Rojo et al. [54,55,2,56,57] investigate ordering of caterpillars, Rojo and Medina [53], and Rojo [51,52] study algebraic connectivity in the class of Bethe trees. Guo et al [24] order lollipop graphs. Liu and Liu [42] order unicyclic graphs with the smallest algebraic connectivity. Li et al. [38]order bicyclic graphs. Wang [67] classifies trees as a function of the algebraic connectivity in certain intervals. Zhang [72] studies algebraic connectivity in trees of diameter 4 and also the relation between diameter and this parameter. Yuan et al. [70] give classes of trees of largest algebraic connectivity.
As an attempt to sort the myriad of results and techniques available in the literature, our purpose in this work is to collect the most important results relative to ordering
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of trees and graphs with few cycles by algebraic connectivity. The types of result we report are, for example, the extremal graphs relative to this parameter; the graph in a particular class that has the largest (smallest) algebraic connectivity or the ordering (partial or total) by algebraic connectivity of graphs in particular classes. We expect the survey will help us finding some theoretical sense of the connectivity of graphs by partial orderings as well as understanding the difficulties of obtaining a total order of graphs (even a total order of trees) by this spectrum invariant.
The paper is organized as follows: in Section 2, we present the basic definitions and notations in order to make this paper self-contained. Section 3 describes the subclass of trees where the ordering by algebraic connectivity is known and discusses the behavior of algebraic connectivity as a function of the diameter. In Section 4, we consider the algebraic connectivity of unicyclic graphs and, in Section 5, we discuss this ordering in classes of more general graphs. Finally, the last section is devoted to conclusions, including a list of research problems to challenge those interested researchers.
2. Preliminaries
Let G = (V, E) be an undirected and simple graph, where V = {v1, . . . , vn} is the set of vertices and E = {{vi, vj} : vi, vj ∈ V } is the set of edges. Let us denote by Kn the complete graph, by Pn the path, by K1,n−1 the star and by Cn the cycle, all of them with n vertices. A tree is a connected and acyclic graph; a unicyclic, a bicyclic and a tricycle graph are connected graphs with n, n +1, and n +2 edges, respectively. The girth of a graph G is the length of the shortest cycle in G and the greatest distance between any two vertices in G is the diameter of G. We denote by δ(G) the minimum degree and Δ(G) the maximum degree.
The adjacency matrix is denoted by A(G) and D(G) is the diagonal matrix in which di,i = d(vi) is the degree of the vertex vi in G. The matrix L(G) = D(G) − A(G) is known as the Laplacian matrix of G. It is well known that L(G) is an n × n positive semidefinite matrix and their eigenvalues are the roots of the characteristic polynomial of L(G), denoted by Φ(G). We set 0 = μ1 ≤ μ2 ≤ . . . ≤ μn as the eigenvalues of L(G). It is well known that the multiplicity of 0 as an eigenvalue of L(G) is the number of connected components of G. This implies that, for connected graphs, the second smallest eigenvalue of G, μ2 �= 0. In 1973, Fiedler [14] published a seminal paper about this invariant and named it the algebraic connectivity of the graph G, denoted by a(G). Since then a(G)has been considered as a metric to measure the connectivity of a graph. Other known connectivity measures are the vertex connectivity κ(G) (for G �= Kn is the minimum number of vertices whose deletion yields a disconnected graph and κ(Kn) = n − 1), and the edge connectivity e(G) (the minimum number of edges whose deletion yields a disconnected graph). Also it’s worth noting that the vertex connectivity for Kn does not follow the definition above.
In that seminal paper, Fiedler proves that a(G) ≤ κ(G) for G �= Kn and, as a consequence, we have that a(G) ≤ e(G) and a(G) ≤ δ(G). He also proves that when an
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edge is removed from a connected graph, the algebraic connectivity does not increase. Moreover, if we remove k vertices from a graph and all adjacent edges, the algebraic connectivity of the resulting graph is at least the algebraic connectivity of the original graph minus k. If G is a complete graph, a(G) = n and, in any other case, a(G) ≤n −2. Another interesting observation is that while the vertex connectivity and the edge connectivity equal 1 for trees, the algebraic connectivity is a positive number no larger than 1, except for K2 where a(K2) = 2. In fact, if T is the star K1,n−1, then a(T ) = 1(see [14]). Fiedler [16] proves that if T is a tree on n vertices, then 2(1 − cos(πn )) ≤ a(T ); the equality holds if and only if T = Pn. Grone and Merris [25], in 1990, show that, if T �= K1,n−1 is a tree on n ≥ 6 vertices then a(T ) < 0.49. From these results, we conclude that, for every tree T and n > 2, a(T ) belongs to the interval (0, 1]. Besides, there is a large gap between the algebraic connectivity of the star (equal to 1) and that of any other tree with n ≥ 6 vertices, which is unable to reach 0.49 (see Abreu 2007 [1]). Kirkland [31] proves that each nonnegative real number is a limit point for algebraic connectivity, and characterizes the limit points for algebraic connectivity of trees.
It is well known that Pn is the tree of order n with minimum algebraic connectivity. Fiedler [14] shows that the algebraic connectivity of a spanning subgraph of G, cannotbe larger than the algebraic connectivity of the original graph. Using this and the fact that a connected graph G with n vertices has a spanning tree, it follows that
a(Pn) = 2(
1 − cos(π
n
))≤ a(G).
One of the most important tools that Fiedler has put forward in his papers is what is now known as the Fiedler vector which is an eigenvector associated with the algebraic connectivity [15]. Given a Fiedler vector y = (y1, . . . , yn) we may label the vertex vi (the order induced by the Laplacian matrix) with yi. Fiedler shows that this labeling, called a characteristic valuation, gives information about the graph.
In particular, based on the sign of the characteristic valuation, a block decomposition of the graph is obtained, where each block is two-connected. This decomposition meets important applications in partition and clustering problems (see, for example, [10,49,59,61,62]). Applications of the algebraic connectivity outside graph theory (where sometimes it is called spectral gap) can also be seen in [44], where a more analytical point of view of the eigenvectors is given, in [28], where a geometric interpretation of the eigenspace of the Fiedler vector is presented using semidefinite programming duality and in [43], where the connectivity of networks is discussed.
Regarding ordering by algebraic connectivity, the object of this survey paper, we refer to [47], where an account of Fiedler’s influence on spectral graph theory is given. Of par-ticular importance for ordering graphs is Fiedler’s Monotonicity Theorem. A special case of the discrete nodal domain theorem [9], it states there is monotonicity on the charac-teristic valuation along paths going through articulation points. Since the consequences of this result for ordering trees by algebraic connectivity is probably the most important
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tool for the subject, we present these results for trees. The monotonicity theorem of Fiedler [15] reads as follows.
Theorem 1. If y is an eigenvector associated with a(T ) then exactly one of the following two cases occurs:
(A) No entry of y is 0. In this case, there is a unique pair of vertices vi and vj such that vi and vj are adjacent with 0 < yi and yj < 0.
(B) Some entry of y is 0. In this case, the subgraph induced by the vertices corresponding to zeros in y is a connected subgraph. Moreover, there is a unique vertex vk such that yk = 0 and vk is adjacent to a vertex vi with yi �= 0.
Definition 2. (See [45].)
(i) T is called a Type II tree if (A) holds. In this case, the vertices vi and vj are called the characteristic vertices of T .
(ii) T is called a Type I tree if (B) holds. In this case, the vertex vk is called the characteristic vertex of T .
Lemma 3. (See [33].) Let Lk be the principal submatrix of the Laplacian matrix L(T )obtained by deleting the k-row and k-column from L(T ). Then the (i, j)-entry of L−1
k is equal to the number of edges of T which are on both the path from vertex vi to vertex vkand the path from vertex vj to vertex vk.
Hence the (i, j) entry of L−1k is positive if and only if the vertices vi and vj are in the
same branch of T at the vertex vk. Consequently, there is a labeling of the vertices of T such that L−1
k is similar to a block diagonal matrix in which the number of diagonal blocks is the degree of the vertex vk. Moreover, each diagonal block is a positive matrix corresponding to a unique branch at vk. This matrix is called the bottleneck matrix for that branch at vk. From the Perron–Frobenius Theory for nonnegative matrices, it follows that each diagonal block has a simple eigenvalue equal to its spectral radius, its Perron root. A branch is called a Perron branch if the Perron root of its bottleneck matrix is equal to the spectral radius of L−1
k .For a vertex v of a graph G, consider the connected components S1, . . . , S� of G − v
(notice that � ≥ 2 if and only if v is an articulation point) and denote by L(Sk) the principal submatrix of L(G) corresponding to the vertices of Sk. We say that Sk is a Perron component of G at v if its Perron root is the maximum among all the Perron roots of S1, . . . , S�. Kirkland & Fallat [34] show how to compute the algebraic connectivity using the Perron roots.
Let J be the all ones matrix of the appropriate order and let ρ(A) be the spectral radius of a matrix A.
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Theorem 4. (See [33].)
1. T is a Type I tree with characteristic vertex vk if and only if there are two or more Perron branches of T at vk. In this case,
a(T ) = 1ρ(L−1
k ).
2. Let vi and vj be adjacent vertices of T . Then T is a Type II tree with characteristic vertices vi and vj if and only there exists 0 < γ < 1 such that ρ(M − γJ) =ρ(M̂ − (1 − γ)J) where M is the bottleneck matrix of the branch at vj containing viand M̂ is the bottleneck matrix of the branch at vi containing vj. Moreover, if this condition holds then
1a(T ) = ρ(M − γJ) = ρ
(M̂ − (1 − γ)J
).
3. Ordering trees
It is known that over the class of trees with n vertices, the algebraic connectivity is uniquely minimized by the path, and uniquely maximized by the star. In a similar spirit, the unique trees that maximize and minimize the algebraic connectivity over all trees with fixed diameter are completely characterized in [11]. Ordering all trees with equal number of vertices by algebraic connectivity seems to be a harder problem, due the difficulties of technical details to surmount. As it is observed in the literature, the algebraic connectivity and the diameter of trees seem to be reciprocally related. This correlation also reflects the presentation we chose to give here.
Two techniques may be identified in the publications we have came across. We will call them algebraic and structural techniques, respectively. The algebraic technique uses classical tools from matrix theory to analyze and obtain properties of the characteristic polynomial of the trees involved. As examples, papers that use this algebraic approach are [25,55,72], whose results are detailed below. What we denote structural technique is based on the results stated above about the Fiedler vector associated to trees. The papers on caterpillars reported here are based on this technique.
3.1. Trees with diameter 3
The ordering of trees with diameter 3 is given by Grone and Merris [25]. These trees have exactly two articulation points (cut vertices) and they are adjacent. The notation, T (k, l, 3), reflects the fact that one of these vertices is connected to k pendant vertices (leaves) and the other vertex connects to l other leaves, hence n = k + l + 2. Without loss of generality, we may consider 1 ≤ k ≤ l. Grone and Merris give a total ordering of trees in the class T (k, l, 3) as follows.
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Fig. 1. a(T (k, l, 3)) < a(T (k − 1, l + 1, 3)), 1 ≤ k ≤ n−22 .
Theorem 5. (See [25, Corollary 2].) Let n = k+ l+2 be fixed. Then, L(T (k, l, 3)) has ex-actly one positive eigenvalue less than 1, which is the algebraic connectivity, a(T (k, l, 3)). Moreover, ak = a(T (k, l, 3)) is a strictly decreasing function of k, for 1 ≤ k ≤ (n − 2)/2.
This result was proved as a corollary of [25, Proposition 1] which gives the character-istic polynomial of L(T ), for T ∈ T (k, l, 3),
Φ(T ) = x(x− 1)n−4(x3 − (n + 2)x2 + (2n + kl + 1)x− n).
It is easy to prove this result for n = 4 and 5. For n ≥ 6, they analyze the roots of the polynomial
pk = x3 − (n + 2)x2 + (2n + kl + 1)x− n
as a function of k and l; they prove that L(T (k, l, 3)) has exactly one eigenvalue ak =a(T (k, l, 3)) ∈ (0, 1). Since,
pk(x) − pk−1(x) = (l − k + 1)x,
they show that ak < ak−1, 2 ≤ k ≤ n−22 .
Fig. 1 displays 2 trees in the class T (k, l, 3) such that the tree on the left has smaller algebraic connectivity than the other.
The family T (k, l, 3) is one of the few large class of graphs that is totally ordered by algebraic connectivity. In Abreu, 2007 [1] we find “although several results obtained by Grone and Merris provided distinct partial orders by a(T ) on different subclasses of trees, their results are not enough to give us a total ordering on the set of all trees”, building evidence of the difficulty of the ordering problem.
3.2. Caterpillars
A caterpillar is a tree with at least 4 vertices in which the removal of all leaves makes it a path with at least 2 vertices. We noticed that all the diameter 3 trees, discussed in the previous subsection, are caterpillars. Fallat and Kirkland [11] generalize the result given by Grone and Merris [25], where a complete order for trees of diameter 3 is obtained (see Section 3.1). In a similar fashion, Fallat and Kirkland [11] studied the trees T (k, l, d)built from a path on d − 1 vertices by adding k pendant vertices adjacent to vertex 1 and l pendant vertices adjacent to vertex d − 1 in the path. All trees in this class are caterpillars with diameter d also known as double brooms.
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Fig. 2. a(T (k, l, d)) < a(T (k − 1, l + 1, d)).
Fig. 3. Ordering in the class of trees P (k, l, d).
Fig. 4. C(p) ∈ C, pi ≥ 1 ∀i.
The authors proved that a(T (k, l, d)) < a(T (k − 1, l + 1, d)) when 1 ≤ k ≤ �n−d+12 �.
To illustrate this, the tree on the left in Fig. 2 has smaller algebraic connectivity than the tree on the right.
They also define the trees P (k, l, d) with n vertices and diameter d (odd), obtained from a path on d + 1 vertices and adding k pendant vertices to vertex d+1
2 and l =n − k− (d + 1) pendant vertices to vertex d+3
2 . For k, l and d fixed, Fallat and Kirkland [11] prove that a(P (k, l, d)) < a(P (k − 1, l + 1, d)), obtaining a total ordering of trees P (k, l, d) with n vertices and fixed diameter d. Fig. 3 illustrates this ordering.
Those authors show that among all trees with n vertices and diameter d, the tree T (�n−d+1
2 �, �n−d+12 , d) minimizes the algebraic connectivity, whereas the tree obtained
by connecting n − d − 1 pendant vertices to a central vertex of the path Pd+1 maximizes it.
Rojo et al. [55] consider the following subset of caterpillars. For d ≥ 3, n > 2(d − 1)and p = (p1, p2, . . . , pd−1), let
C ={C(p) : p1 + p2 + · · · + pd−1 = n− d + 1, pi ≥ 1, ∀1 ≤ i ≤ d− 1
}a subclass of caterpillars of order n and diameter d. In that paper they give an upper bound for the algebraic connectivity for all C(p) ∈ C (see Fig. 4), according to the following result.
Theorem 6. (See [55, Theorem 3] and [56, Theorem 3.3, Theorem 3.6, Theorem 3.8].) If C(p) ∈ C and d = 2s + 1 or d = 2s + 2, then
a(C(p)
)≤ a
(C(p̃)
),
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Fig. 5. T (n, k, p1, p2, . . . , pk).
where p̃ = (1, ..., 1, ps+1, 1, ..., 1) and ps+1 = n − 2d + 3. Among all trees in C, C(p̃) and C(p̂) have largest and smallest algebraic connectivity, respectively, where p̂ = (�n−2d+4
2 �, 1, . . . , 1, �n−2d+42 ).
Several ordering results on subclasses of C may be found in [55–57] where we find the ordering of symmetric caterpillars. Theorem 1 in [4] gives a result about the minimal algebraic connectivity of trees with given number of vertices and degree sequence. The authors showed that this extremal value is attended by a caterpillar and also proved conditions about the degrees of the non-pendant vertices.
3.3. Trees with diameter 4
Important contributions to the problem in the class of trees with n vertices and diameter 4 are due to X. D. Zhang [72]. In order to state some of the main results of this author, we review some notation. Let T (n, k, p1, p2, ...., pk) be the tree of order nobtained from the stars K1,k, K1,p1 , K1,p2 , . . . , K1,pk
by identifying the pendant vertices of K1,k with the roots of K1,p1 , K1,p2 , . . . , K1,pk
, respectively. Assuming k ≥ 2 and 0 < p2 ≤ p1, assures that T (n, k, p1, p2, . . . , pk) is a tree of diameter 4 and by allowing 0 ≤ pk ≤ pk−1 ≤ · · · ≤ p1 ensures that T (n, k, p1, p2, . . . , pk) may have pendant vertices at the center vertex of K1,k. We notice that n = p1 + p2 + · · · + pk + k + 1 is the total number of vertices. Conversely, any tree of order n and diameter 4 can be expressed in the form T (n, k, p1, p2, . . . , pk), with 0 < p2 ≤ p1, pk ≥ 0, for k > 2. We illustrate this construction of the general diameter 4 tree in Fig. 5. For brevity, we denote the algebraic connectivity of T (n, k, p1, p2, . . . , pk) by a(p1, p2, . . . , pk).
An important result in [72] is given in the following theorem.
Theorem 7. (See [72, Lemma 2.1].) The algebraic connectivity of T (n, k, p1, p2, . . . , pk)satisfies the inequalities
12
(p1 + 2 −
√p21 + 4p1
)≤ a(p1, p2 . . . , pk) ≤
12
(p2 + 2 −
√p22 + 4p2
).
Equality holds if and only if p1 = p2.
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Zhang also shows [72, Corollary 2.2] that among all trees of order n with diame-ter 4, the tree T (n, n − 3, 1, 1, 0, . . . , 0) has the largest algebraic connectivity. Hence a(T (n, k, p1, p2, . . . , pk)) ≤ a(T (n, n − 3, 1, 1, 0, . . . , 0)) = 3−
√5
2 . For n ≥ 6 and d = 4there is no tree T with 0.381966 � 3−
√5
2 ≤ a(T ) ≤ 0.49 (see [26]). Moreover, it is proved [72, Corollary 2.3] that if p1 = p2 then T (n, k, p1, p1, p3, . . . , pk) is a Type I tree and if p2 < p1 then T (n, k, p1, p2, p3, . . . , pk) is a Type II tree.
Let F1 be the subclass of Type I trees with diameter 4, that is,
F1 ={T (n, k, p1, p2, . . . , pk) :
n =k∑
i=1pi + k + 1, 2 ≤ k, 1 ≤ p1 = p2, and pk ≤ . . . ≤ p3 ≤ p2 = p1
}.
From the fact that the function c(x) = x + 2 −√x2 + 4x is strictly decreasing function,
we may conclude the following result.
Theorem 8. Let T1 = T (n, k, p1, p2, . . . , pk) ∈ F1 and T2 = T (n, k, q1, q2, . . . , qk) ∈ F1. If p1 = q1 then a(T1) = a(T2) and if p1 < q1 then a(T2) < a(T1).
Hence, the theorem above gives an ordering in F1 in the sense that the algebraic connectivity of any two trees can compared without actually computing them.
3.4. Algebraic connectivity versus diameter
In 1990, Grone, Merris and Sunder [26] give a useful upper bound for the algebraic connectivity of trees relating to its diameter.
Theorem 9. (See [26, Corollary 4.4].) Let T be a tree on n vertices with diameter d ≤n − 1. Then,
a(T ) ≤ 2(
1 − cos(
π
d + 1
)).
Another result that relates reciprocally the diameter and the algebraic connectivity is given by Fallat and Kirkland [11] who study the behavior of the algebraic connectivity for trees of the form T (�n−d+1
2 ��n−d+12 , d) with fixed number of vertices n when the
diameter d increases.
Theorem 10. (See [11, Lemma 3.5].) Fix n and for d ≥ 3, 1 ≤ k ≤ l with n = k + l +d − 1, consider T (k, l, d) the tree of diameter d. Let μd be the algebraic connectivity of T (�n−d+1�, �n−d+1, d). Then μd is a strictly decreasing function of d.
2 2
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Fig. 6. From Table 1 of [25].
Fig. 7. From Table 2 of [25].
In Godsil and Royle [19], we find “it has been noted empirically that a(G) seems to give a fairly natural measure of the ‘shape’. Graphs with small values of a(G) tend to be elongated graphs of large diameter with bridges whereas graphs with larger values of a(G)tend to rounder with smaller diameter, and larger girth and connectivity”. The results of [5] may be seen as a step towards a concretization of such a statement, proving that graphs with prescribed order and size that have minimal algebraic connectivity must consist of a chain of cliques. According to Mohar [46], McKay, in a private unpublished communication, proved that if G has n vertices and diameter d, then 4
nd ≤ a(G). This lower bound is close to optimal for trees and its proof along with other stronger bound may be found in [46].
The values of algebraic connectivity for the three trees in Fig. 6 are 0.382 ≈ a(T4) <0.466 ≈ a(T2) < a(T1) = 1, and their diameters, respectively, are diam(T1) = 2 <diam(T2) = 3 < diam(T4) = 4 (see [25]).
These facts above seem to give evidence that a(T ) decreases as the diam(T ) in-creases. However this is not always true. Grone and Merris [25] show that, in some cases, the diameter and algebraic connectivity of trees both increase. Fig. 7 illus-trates this situation. While a(T13) ≈ 0.1864 < a(T12) ≈ 0.1981, the diameters are diam(T13) = 5 < diam(T12) = 6.
More recently, Zhang [72] highlights Grone and Merris’s observation proving that for some trees of diameter 3 there is a tree with sufficiently large diameter with larger algebraic connectivity.
Theorem 11. (See [72].) For any even number n, there is a tree T of order n and diameter d (even) with d <
√n− 1 such that a(T ) > a(T (n2 − 1, n2 − 1, 3)).
The tree T of Zhang is built as follows. Given n and d even numbers, start with the path Pd+1 and s trees of order n1, . . . , ns, respectively, with n1 + · · ·+ns = n −d −1 and then add one edge from a vertex of each of these s trees to the central vertex of Pd+1. He shows that, if the diameter of T is at most
√n− 1 then a(T ) > a(T (n − 1, n − 1, 3)).
2 2
440 N. Abreu et al. / Linear Algebra and its Applications 458 (2014) 429–453
Fig. 8. Tk,l and Tk−1,l+1.
From Zhang’s result, we see that there are many trees with large diameter and large al-gebraic connectivity, indeed larger than the algebraic connectivity of trees of diameter 3. For instance, if n = 64, we have that a(T (31, 31, 3)) ≈ 0.058926 (the tree of diameter 3 and 64 vertices of smallest algebraic connectivity). In this case, consider
√64 = 8, √
64− 1 = 7, and d = 6 (an even number d < 7). Then the tree T64 of order n = 64 with diameter 6 can be constructed from a path P7 by attaching 57 pendant vertices at the third vertex; it can be verified that a(T (31, 31, 3)) < a(T64) = a(P7) ≈ 0.198062.
To end this section it is worthwhile to point out the bound a(T ) < 2 −√
3 for trees with n ≥ 7 and d ≥ 5 given by Yuan, Shao and Zhang [70]. We notice that this bound does not hold by the path P5 (with diameter 4). We also observe that the tree T64 presented above verifies the bound given by [70] a(T64) = a(P7) ≈ 0.198062 < 2 −
√3 ≈ 0.26795.
3.5. Results on other trees
In 2008, Patra and Lal [48] define the grafting operation as follows. Let T be a tree on n vertices and v a vertex of T . For k ≥ l ≥ 1 let Tk,l be the graph obtained from Tby attaching at v two new paths, P : vv1v2 . . . vl and Q : vu1u2 . . . uk of length l and k, respectively. Also, let Tk−1,l+1 be the graph obtained from Tk,l by removing the edge {uk−1, uk} and adding the edge {vl, vl+1}. Fig. 8 displays the trees Tk,l and Tk−1,l+1.
Using the definition above, the authors compare the algebraic connectivity among all pairs of trees Tk,l and Tk−1,l+1 and give the following result.
Theorem 12. (See [48, Theorem 2.4]) Let T be a tree on n ≥ 2 vertices and v a vertex of T . Let Tk,l be the graph defined before. If l ≥ k ≥ 1 then a(Tk,l) ≥ a(Tk−1,l+1).
In the same year Yuan, Shao and Zhang [70] describe six classes of trees with the greatest algebraic connectivity for fixed number of vertices. To obtain their result, the trees Tk,p,q with n = 3k + 2p + q + 1, where there is a vertex v of degree k + p + q and such that Tk,p,q − v = kK1,2 ∪ pK1,1 ∪ qK1 (where it means a disjoint union of k copies of K1,2, p copies of K1,1 and q copies of K1) are introduced (see Fig. 9).
N. Abreu et al. / Linear Algebra and its Applications 458 (2014) 429–453 441
Fig. 9. Tk,p,q.
The following six classes of trees of order n are defined as follows: C1 = {T0,0,n−1}(the star K1, n− 1); C2 = {T0,1,n−3} (the quasi-star with n vertices, T (1, n − 3, 3));C3 = {T0,p,q : p ≥ 2, 2p + q = n − 1}; C4 = {T1,0,n−4} (the tree T (2, n − 4, 3)); C5 ={T1,p,q : p ≥ 1, 2p + q = n − 4} and C6 = {Tk,p,q : k ≥ 2, 3k + 2p + q = n − 1}.
By using Perron components and characteristic polynomials, the authors prove the next result.
Theorem 13. Let T be a tree with order n and diameter d.
(a) If n ≥ 7 and d ≥ 5 then a(T ) < 2 −√
3;(b) For n ≥ 15, 2 −
√3 ≤ a(T ) if and only if T ∈
⋃6i=1 Ci [70, Theorem 3.6].
(c) If 1 ≤ i < j ≤ 6 and Ti ∈ Ci and Tj ∈ Cj then a(Tj) < a(Ti).
In [67], all trees of order n ≥ 45 with algebraic connectivity in the interval [ 5−√
212 ,
2 −√
3) are determined.
4. Ordering unicyclic graphs
Let Un, n ≥ 3 be the family of all unicyclic graphs of order n. Let Pn,k be the graph with n vertices obtained by connecting k pendant vertices to any vertex of Kn−k. Let Cn,g be the lollipop, obtained by appending a cycle Cg to a pendant vertex of a path Pn−g. The extremal graphs on Un are determined by Lal, Patra and Sahoo in 2011 [35], as follows.
Theorem 14. (See [35, Theorem 8, Theorem 9].) Let n ≥ 3 be fixed.
(a) Maximum algebraic connectivity in Un is uniquely attained by Cn if n ≤ 5, by Pn−3n
if n > 6. When n = 6, C6 and P 36 are the only two graphs, up to isomorphism,
having the maximum algebraic connectivity over U6.(b) Minimum algebraic connectivity in Un is uniquely attained by Cn,3
Since the algebraic connectivity is monotone on spanning subgraphs [14], it is clear that among all connected graphs with n vertices and specified girth g (the length of the shortest cycle), the minimum algebraic connectivity occurs for a unicyclic graph. Let Un,g be the class of connected graphs on n vertices and a unique cycle of length g.
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Fig. 10. Pn−3n .
Fig. 11. Gg:n1,n2,...,ngand Gg,n−g.
This class has been considered by Fallat, Kirkland in [11] and Fallat, Kirkland and Pati in [12,13]. In the first paper it is conjectured that the minimum algebraic connectivity in Un,g is given by a(Cn,g).The conjecture for g = 3, g = 4 and also n > 3g− 1 is proved in [11,12]. Later Guo [22] removes this condition and completes the proof.
Theorem 15. (See [12, Theorem 3.21] and [22, Theorem 2.10].) Among all connected graphs on n vertices and girth g, the algebraic connectivity is minimized by the lollipop Cn,g.
An ordering of lollipop graphs with fixed number of vertices is proved by Fallat et al. [12]. They prove that the algebraic connectivity of Cn,g with fixed n is a non increasing function of the girth g, when n ≥ 3g−1
2 . Again, Guo [22] removes this condition, giving the total order for the lollipop graphs.
Theorem 16. (See [12, Lemma 2.8 ] and [22, Theorem 2.11].) Fix n and suppose that g ≥ 4. Then a(Cn,g−1) < a(Cn,g).
Determining the graph with the largest algebraic connectivity in the class Un,g seems to be a harder problem. Fallat and Kirkland [11] prove that the maximum algebraic connectivity in Un,3 is attained by Pn−3
n , see Fig. 10. We notice that, according to Theorem 14 (a) Pn−3
n also maximizes the algebraic connectivity in Un for n > 6.In order to find maximizers of algebraic connectivity in the class Un,g, Fallat et al. [13]
introduce the following notation. Let Gg:n1,n2,...,ngbe the unicyclic graph in Un,g such
that the vertices of the cycle are labeled 1, . . . , g as we transverse the cycle. For each 1 ≤ i ≤ g, the vertex i has ni pendant vertices and n − g = n1 + . . .+ng. For simplicity, the graph Gg:n−g,0,...,0 is denoted by Gg,n−g. Fig. 11 displays Gg:n1,n2,...,ng
and Gg,n−g.
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Fig. 12. Gg:0,...,0,k and Gg:1,0,...,0,1,k−2.
In the same paper, it is proven that G4,n−4 is the unique graph that maximizes the algebraic connectivity in Un,4. Moreover for n ≥ 5, a(G4,n−4) > 2 −
√2.
Under some conditions of n and g, the next theorem finds the maximizer to the algebraic connectivity in Un,g.
Theorem 17. (See [13, Theorem 10].) Fix g ≥ 5. Then there exists an N such that if n > N , then Gg,n−g is the unique maximizer of algebraic connectivity over the graphs in Un,g.
The theorem above establishes that when n is large relative to g, then Gg,n−g max-imizes algebraic connectivity over Un,g. In [13], the authors require n to be quite a bit larger than g, their computations show that the lower bound on N is roughly g7
π4 for large values of g. On the other hand, the next result shows that when g is large relative to n − g, the graph Gg,n−g does not maximize algebraic connectivity over Un,g.
Theorem 18. (See [13, Theorem 19].) Let n = g + k and k ≥ 3, let G1 = Gg:0,...,0,k and G2 = Gg:1,0,...,0,1,k−2. Then, for fixed n and g sufficiently large a(G1) < a(G2).
Fig. 12 shows the graphs G1 = Gg:0,...,0,k and G2 = Gg:1,0,...,0,1,k−2 in the last theorem.Fallat et al., in the same paper, conjecture that the graph obtained by adding a single
pendant vertex to k consecutive vertices on the cycle is the maximizer for all sufficiently large g.
Back to minimizing algebraic connectivity, it is worth mentioning some results about the graphs with the smallest values of algebraic connectivity in Un. The first seven unicyclic graphs according to their smallest algebraic connectivities were determined by Guo [22] (the first one), Liu and Liu in [42] (the second and the third smallest unicyclic graphs) and Li, Guo and Shiu in [37] (the other ones, for n ≥ 13). The six smallest (by algebraic connectivity) unicyclic graphs can be seen later among all the sixteen smallest graphs with n ≥ 13 in Fig. 15 as W3, W5, W8, W9, W12 and W16.
Unicyclic caterpillars are graphs where the removal of all pendant vertices makes it a cycle. The class of unicyclic caterpillars with n vertices and girth g is denoted by UCn,g. We finish this section reporting two results about them. Observe that the graphs Gg:n1,n2,...,ng
defined above are included in UCn,g. According to Shiu et al. [60] the graph which minimizes the algebraic connectivity in UCn,g is Gg:n1,n2,...,ng
with n1 = �n−g2 �
and n� g �+1 = �n−g and nj = 0 ∀j �= 1, � g � + 1.
2 2 2
444 N. Abreu et al. / Linear Algebra and its Applications 458 (2014) 429–453
Fig. 13. Gg:n11,n
12,...,n
1g
and Gg:n21,n
22,...,n
2g.
Fig. 14. Operation.
In order to obtain the minimizer of algebraic connectivity in UCn,g, the authors find a partial order of a unicyclic caterpillar subclass (see Fig. 13).
Theorem 19. (See [60, Lemma 2.16].) For n1 ≥ n2+2, a(Gg:n11,n
12,...,n
1g) > a(Gg:n2
1,n22,...,n
2g)
where n21 = n1
1 − 1, n2� g
2 �+1 = n1� g
2 �+1 + 1 and ni = 0 for all i �= 1, � g2� + 1.
5. Ordering more general connected graphs
Shao, Guo and Shan, in 2008 [58] study the operation illustrated in Fig. 14 and obtain the following.
Corollary 20. (See [58, Corollary 4.2].) Let G be a connected graph on n vertices. Suppose that v1, . . . , vs, s ≥ 2 are s pendant vertices of G adjacent to a common vertex v. Let G̃be a graph obtained from G by adding any t, 0 ≤ t ≤ s(s−1)
2 edges among v1, . . . , vs. If a(G) �= 1, then a(G̃) = a(G).
In the same paper, the authors determined the connected graphs of order n ≥ 9 with smallest algebraic connectivity.
Theorem 21. (See [58].) Let W1, W2, W3, W4, W5, W6, W7 be the connected graphs of order n ≥ 9 given in Fig. 15. Then a(W1) < a(W2) = a(W3) < a(W4) = a(W5) =a(W6) < a(W7); and, for any connected graph G of order n with G /∈ {W1, . . . , W6} we have a(W7) ≤ a(G).
Let Bn be the set of bicyclic graphs of order n, in the paper [38], Li et al. determine four bicyclic graphs among all graphs in Bn with n ≥ 13 with smallest algebraic connectivity.
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Fig. 15. 16 graphs with smallest value of algebraic connectivity and n ≥ 13.
Their result together with results relative to trees and unicyclic graphs are used to determine the smallest sixteen graphs among all connected graphs of order n, which extends the results given by Shao et al. in [58]. See Fig. 15 where the four bicyclic graphs in Bn with n ≥ 13 are W6, W10, W13 and the bicyclic graph obtained by adding the edge (x, y) in W16.
Guo, in 2010 [23], studying the behavior of the algebraic connectivity under grafting and edge separation, obtained the following generalization of Lemma 2.10 of Fallat, Kirkland and Pati [12] and Theorem 2.4 of Patra and Lal [48].
Theorem 22. (See [23, Theorem 4.4].) Let G be a connected graph with at least two vertices and let Gk,l and Gk+1,l−1, k ≥ l ≥ 1 be the graphs in Fig. 16. Let X be a Fiedler vector of Gk,l. Then a(Gk+1,l−1) ≤ a(Gk,l) and the inequality is strict if either X(v1) �= 0 or X(u1) �= 0.
446 N. Abreu et al. / Linear Algebra and its Applications 458 (2014) 429–453
Fig. 16. Gk,l and Gk+1,l−1.
Fig. 17. Pkn , k = n − 2 and Pn,n−2.
Given integers n and k with 0 ≤ k ≤ n, a pineapple, denoted by P kn , is a graph with
n vertices consisting of a clique on n − k vertices and a stable set on the remaining kvertices, in which each vertex of the stable set is adjacent to a unique and the same vertex of the clique. Lal, Patra and Sahoo in 2010 [35] study the algebraic connectivity of connected graphs with fixed order n and k pendant vertices, denoted by Hn,k. In this class the pineapples P k
n for k �= n − 2 and the graphs Pn,n−2 given in Fig. 17 play an important role.
Theorem 23. (See [35, Theorem 2, Theorem 3].) P kn for k = 1, . . . , n − 1, uniquely
maximizes the algebraic connectivity over Hn,k.
As for the minimum value of the algebraic connectivity, consider the trees T (n, l, d)as illustrated in Fig. 2.
Theorem 24. (See [35, Theorem 5].) The tree T (�k2 , �
k2 �, n − k) uniquely attains the
minimum algebraic connectivity over Hn,k, k ≥ 2.
The same authors consider the subclass Tn,k of all trees in Hn,k. For 2 ≤ k ≤ n −1, the tree T̃n,k of order n such that there is a vertex v of degree k and T̃n,k−v = rPq+1∪(k−r)Pq
where q = �n−1k � and r = n − 1 − kq has maximum algebraic connectivity in Tn,k [35,
Theorem 6]. As we can see in Fig. 18, the graph T̃n,k − v is the disjoint union of r copies of Pq+1 and q − r copies of Pq. If n = kq + 1, then r = 0, and as an example we have T̃4,3.
The same paper also considered Fn, n ≥ 3, the class of all connected graphs of order n without any pendant vertex. For n ≥ 6, the graph Cn−6
3,3 , a path on n − 6 vertices with
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Fig. 18. T̃n,k, q = �n−1k � and r = n − 1 − kq.
Fig. 19. Cn−63,3 and C0
3,3.
a 3-cycle added to each of its two pendant vertices (see Fig. 19), attains the minimum algebraic connectivity over Fn.
6. Final remarks
There seems to be many open problems on ordering graphs by means of any parameter. Perhaps finding a parameter that can distinguish graphs apart from each other is still a challenging task. Whether algebraic connectivity is such a good/bad parameter is yet to be proven.
If one can find large sets of n-vertex (essentially distinct) graphs that have the same algebraic connectivity, then one may argue that it is a bad parameter. For example, since Corollary 20 provides a simple method for creating graphs with the same algebraic connectivity, one may present counting arguments to partially answer some of these questions.
Turning to a more experimental line of investigation, we refer to an interesting page created by João Carvalho,1 where the catalogue of trees of Brendan McKay2 has been used. With all trees up to 22 vertices, this page enables one to experiment with sev-eral spectral parameters of the adjacency, the Laplacian and the normalized Laplacian matrices. Using that data set one finds, for example, that among all 235 trees with 11 vertices, there are 5 pairs, 1 triplet, 3 quadruples and 1 sextuple that have equal algebraic connectivity. Therefore there are 21 trees on 11 vertices that cannot be distinguished by algebraic connectivity. This percentage of undistinguishable trees seems to decrease as the number of vertices grows larger. Since these are numerical experiments, there may be errors involved, but they may help formulating conjectures. Therefore, as suggested problem, one may prove (or disprove) that when n gets larger, the number of trees on n vertices that are undistinguishable by algebraic connectivity gets smaller. Fig. 20 dis-plays trees with the same order, diameter and algebraic connectivity. These trees can be
1 http://www2.mat.ufrgs.br/~carvalho/pesquisa/interativa/graphenergy/graphspec.php.2 http://cs.anu.edu.au/~bdm/data/trees.html.
448 N. Abreu et al. / Linear Algebra and its Applications 458 (2014) 429–453
Fig. 20. Two trees with n = 11, d = 4 and algebraic connectivity 0.381966011250105.
found in João Carvalho’s page and have the reference numbers 230 and 231, choosing the following options: n = 11, list of eigenvalues and energy, Laplacian matrix and order by second smallest eigenvalue.
Another useful tool is AGX, available through the page http://www.gerad.ca/~agx/index.php. This is an interactive software designed to help finding conjectures in graph theory. Some of the conjectures that were formulated using AGX, including those involv-ing algebraic connectivity and other invariants, may be found in a database. Recently, one of these conjectures, namely, that for graphs of diameter d, 3 ≤ a(G) + d, has been proved in [8].
Regarding theoretical aspects of tree ordering by algebraic connectivity, it appears that Type I trees are easier to describe than Type II . Kirkland, in [32], shows how to build Type I trees with nonisomorphic Perron branches that have the same algebraic connectivity. It may be worthwhile investigating the algebraic connectivity of trees that are symmetric with respect to characteristic vertices.
Since there is a good knowledge about tree ordering by algebraic connectivity, one might ask whether there exist adding edge operations that preserve these orderings. We notice, for example, that adding an edge between the end points of the largest path of the d-diameter tree with largest algebraic connectivity, transforms it into the d-unicyclic graph with largest algebraic connectivity (see Section 3.2 and Theorem 17 of Section 5). Hence, by adding edges to a class of trees, it may be possible to obtain an order in classes of unicyclic graphs from the induced order (by algebraic connectivity) of the trees.
In the remainder of this section we formulate questions whose answers would shed some light on the relation between the diameter of a tree and its algebraic connectiv-ity.
6.1. Survey questions
The questions listed here are related to the fact that the spectral graph theory com-munity believes that there is a strong connection between the algebraic connectivity of an n-vertex tree and its diameter. More precisely, researchers believe that, if T1 and T2
are n-vertex trees whose diameters satisfy diam(T1) < diam(T2), then typically their algebraic connectivities satisfy a(T2) < a(T1). On the other hand, it is well-known that this relation does not hold in general, i.e. that the set
{(T1, T2) : |T1| = |T2|, diam(T1) < diam(T2) and a(T1) < a(T2)
}
N. Abreu et al. / Linear Algebra and its Applications 458 (2014) 429–453 449
Fig. 21. Two trees with the same spectrum and different diameters.
is nonempty (actually, it is known that this set is infinite). In fact, trees with the same algebraic connectivity (or even the same Laplacian spectrum) may have different diam-eters. We refer to Fig. 21 for an example of two co-spectral trees of diameters 7 and 8, respectively, both with 11 vertices. Related to this line of thought, one might inves-tigate what are the effects on the diameter of tree operations that preserve algebraic connectivity (or that preserve the whole spectrum).
The following two questions might be able to measure the connection between al-gebraic connectivity and diameter. Given a positive integer n, let Tn be the set of all unlabeled trees on n vertices.
In the first question, we wish to express that algebraic connectivity and diameter are related by saying that, as long as the diameters of two n-vertex trees are sufficiently far apart, we can tell which tree has the largest algebraic connectivity. To this end, given n ∈ N , we wish to find a positive integer k = k(n) with the following property: every pair of trees T1, T2 ∈ Tn such that k ≤ diam(T1) − diam(T2) satisfies a(T2) < a(T1). Formally, we let
Sn ={k ∈ N : ∀T1, T2 ∈ Tn,
[k ≤ diam(T2) − diam(T1) ⇒ a(T2) < a(T1)
]}and we are interested in
k(n) = minSn.
Recall that for n ≥ 3 there is a single tree on n vertices and diameter two (the smallest possible diameter), namely the star K1,n−1, and a single tree on n vertices and diameter n − 1 (the largest possible diameter), namely the path Pn. Since a(Pn) < a(K1,n−1), we know that n −3 ∈ Sn. In particular, we have Sn �= ∅ and the function k(n) is well-defined. On the other hand, whenever n is even Zhang [72] has constructed a pair of n-vertex trees such that T1 has diameter three and T2 has diameter d = �√n− 1�, but a(T1) < a(T2). This implies that �√n− 1� − 3 < k(n) for every even value of n.
Question 1. Determine the behavior of function k(n). For instance, is there a continuous function f : R → R such that limn→∞
k(n) = 1?
f(n)
450 N. Abreu et al. / Linear Algebra and its Applications 458 (2014) 429–453
A second approach for comparing the algebraic connectivity and diameter of trees is probabilistic. Here, the intuition is that it is hard to find a pair {T1, T2} such that diam(T1) < diam(T2) and a(T1) < a(T2). To this end, consider the set
Bn ={(T1, T2) : |T1| = |T2| = n,diam(T1) < diam(T2) and a(T1) < a(T2)
}of “bad” pairs, and let
An ={(T1, T2) : |T1| = |T2| = n,diam(T1) < diam(T2)
}denote the set of all possible pairs of trees considered.
Question 2. Determine whether Bn is substantially smaller than An. For instance, is it true that
limn→∞
Bn
An= 0?
Acknowledgements
Oscar Rojo is partially supported by FONDECYT Regular 1100072 and FONDE-CYT Regular 1130135, Chile. Vilmar Trevisan is partially supported by CNPq – Grants 305583/2012-3 and 481551/2012-3 and by CAPES Grant PROBRAL 408/13 – Brazil. Nair Abreu and Claudia Justel are partially supported by CNPq with Grants 300563/94-9 and 305516/2010-8, respectively. The authors are grateful to Carlos Hoppen for helping us to precisely define a probabilistic version of the problem of relating the algebraic connectivity and diameter and to João Carvalho, who set up the webpage we can experiment with.
Finally, we would like to acknowledge the careful reading of the referees and the handling editor whose suggestions greatly improved the readability of this survey.
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