conjecture for the geometric-arithmetic index with given minimum

Conference in honour of prof. Dragoš Cvetkovic 2016

CONJECTURE

FOR THE

GEOMETRIC-ARITHMETIC

INDEX WITH GIVEN

MINIMUM DEGREE

Lj. Pavlovica, M. Milivojevica

aFaculty of Science, University of Kragujevac, Serbia

Beograd, 2016 Conference 2016 – p. 1/35

Introduction to the Geometric-Arithmetic index

◮ Abstract. The geometric-arithmetic indexGA(G) of a graphis defined as sum of weights of all edges of graph. The weightof one edge is quotient of the geometric and arithmetic mean of

degrees of its end vertices2√dudv

du+dv. The predictive power of GA

for physico-chemical properties is somewhat better than thepredictive power of other connectivity indices. LetG(k, n) bethe set of connected simple n-vertex graphs with minimumvertex degreek. We give a conjecture about lower bounds andstructure of extremal graphs of this index forn-vertex graphswith given minimum degreek.

Conference 2016 – p. 2/35


◮ Abstract. The geometric-arithmetic indexGA(G) of a graphis defined as sum of weights of all edges of graph. The weightof one edge is quotient of the geometric and arithmetic mean of

degrees of its end vertices2√dudv

du+dv. The predictive power of GA

for physico-chemical properties is somewhat better than thepredictive power of other connectivity indices. LetG(k, n) bethe set of connected simple n-vertex graphs with minimumvertex degreek. We give a conjecture about lower bounds andstructure of extremal graphs of this index forn-vertex graphswith given minimum degreek.

◮ Keywords: Geometric-arithmetic index, Linear programming.



◮ G = G(k, n) – simple connectedn-vertex graphs withδ(G) = k.




◮ u - a vertex ofG, d(u) - the degree of this vertex.(uv) -an edgewhose endpoints are the verticesu andv.





17 D. Vukicevic, B. Furtula,Topological index based on theratios of geometrical and arithmetical means of end -vertexdegrees of edges, Journal of Mathematical Chemistry, Vol. 46,Issue 2 (2009), 1369-1376.






◮ The Geometric-Arithmetic is:

GA(G) =∑

uv∈E(G)

2√dudv

du + dv,

where the summation goes over all edgesuv of G.Conference 2016 – p. 3/35


◮

t��

HHHH

HH

��

@@@@

######

cccc

cc

AAAAA

��

s

s

s

s

ss

ss

4

22

1 1

6

1

2

3

Fig. 1. GA(G) = 22√1·45

+ 2√1·67

+ 2√2·24

+ 2√2·35

+32√2·68

+ 2√3·47

+ 2√3·69

+ 2√4·6

10



◮ In fact, this index belongs to wider class of so-calledgeometric-arithmetic general topological indices. A class ofgeometric-arithmetic general topological indices is defined in[9]

GAgeneral(G) =∑

uv∈E(G)

2√QuQv

Qu +Qv

,

whereQu is some quantity that (in a unique manner) can beassociated with the vertexu of the graphG.

It is easy to recognize thatGA is the first representative of thisclass obtained by settingQu = du.



◮ In fact, this index belongs to wider class of so-calledgeometric-arithmetic general topological indices. A class ofgeometric-arithmetic general topological indices is defined in[9]

GAgeneral(G) =∑

uv∈E(G)

2√QuQv

Qu +Qv

,

whereQu is some quantity that (in a unique manner) can beassociated with the vertexu of the graphG.

It is easy to recognize thatGA is the first representative of thisclass obtained by settingQu = du.

◮ [9] G. Fath-Tabar, B. Furtula, I. Gutman,A newgeometric-arithmetic index, Journal of MathematicalChemistry, Vol. 47, Issue 1 (2010), 477-486.



◮ The second member of this class was considered by Fath-Tabaret al. [9] by settingQu to be the numbernu of vertices ofGlying closer to the vertexu than to the vertexv for the edgeuvof the graphG:

GA2(G) =∑

uv∈E(G)

2√nunv

nu + nv

.




GA2(G) =∑

uv∈E(G)

2√nunv

nu + nv

.

◮ In [9] the main properties ofGA2 were established, includinglower and upper bounds.




GA2(G) =∑

uv∈E(G)

2√nunv

nu + nv

.

◮ In [9] the main properties ofGA2 were established, includinglower and upper bounds.

◮ Zhou et al. [19] proposed a third member of the class ofGAgeneral by settingQu to be the numbermu of the edges ofG, lying closer to vertexu than to vertexv.


Known results for the Geometric-Arithmetic index

◮ It is noted in [17] that the predictive power ofGA forphysico-chemical properties ( boiling point, entropy, enthalpyand standard enthalpy of vaporisation, enthalpy of formation,acentric factor ) is somewhat better than the predictive powerof the Randic connectivity index.




◮ In [17] Vukicevic and Furtula gave the lower and upper boundsfor theGA, identified the trees with the minimum and themaximumGA indices, which are the star and the pathrespectively.




◮ In [17] Vukicevic and Furtula gave the lower and upper boundsfor theGA, identified the trees with the minimum and themaximumGA indices, which are the star and the pathrespectively.

◮ In [18] Yuan, Yhou and Trinajsic gave the lower and upperbounds forGA index of molecular graphs using the numbers ofvertices and edges. They also determined then-vertexmolecular trees with the first, second and third minimum andmaximumGA indices. Conference 2016 – p. 7/35


◮ The Randic connectivity index was studied by chemists andmathematicians and there are a lot of papers about it. Severalbooks are devoted to the Randic index. Recently, thegeometric-arithmetic index attracted attention ofmathematicians also, but there are few papers about it,dedicated to molecular graphs ([8], [12]).




◮ In [5],K. Das, I. Gutman, B. Furtula,Survey on Geometric -Arithmetic Indices of Graphs, MATCH-Communications inMathematical and in Computer Chemistry, 65 (2011), 595-644,authors are collected all obtained results on class GA indices.




◮ In [5],K. Das, I. Gutman, B. Furtula,Survey on Geometric -Arithmetic Indices of Graphs, MATCH-Communications inMathematical and in Computer Chemistry, 65 (2011), 595-644,authors are collected all obtained results on class GA indices.

◮ In [6] T. Divnic, M. Milivojevi c, Lj. Pavlovic,Extremalgraphs for the geometric-arithmetic index with givenminimum degree, Discrete Applied Mathematics, Vol. 162,2014, 386-390, authors found extremal graphs for GA indexfor some given minimum degree. Conference 2016 – p. 8/35

Conjecture about the extremal graphs

◮ Denote byft(k) =(n−t)(k−t)

2+

2√

k(n−t)

k+n−tt(n− t) for 0 ≤ t ≤ k,

k ≤ k0 and bykt ∈ [0, k0] a unique root of equationft+1 − ft = 0.



◮ Denote byft(k) =(n−t)(k−t)

2+

2√

k(n−t)

k+n−tt(n− t) for 0 ≤ t ≤ k,

k ≤ k0 and bykt ∈ [0, k0] a unique root of equationft+1 − ft = 0.

◮ Conjecture. If G is a connected simple n vertex graphswith minimum vertex degree k, then


Conjecture about the extremal graph

◮ GA ≥

kn2, ⌈k0⌉ ≤ k

(n−1)(k−1)2

+2√

k(n−1)

k+n−1(n− 1), ⌈k1⌉ ≤ k ≤ ⌊k0⌋,

(n−2)(k−2)2

+2√

k(n−2)

k+n−22(n− 2), ⌈k2⌉ ≤ k ≤ ⌊k1⌋,

(n−3)(k−3)2

+2√

k(n−3)

k+n−33(n− 3), ⌈k3⌉ ≤ k ≤ ⌊k2⌋,

......

(n−t)(k−t)2

+2√

k(n−t)

k+n−tt(n− t), ⌈kt⌉ ≤ k ≤ ⌊kt−1⌋,

......

2√

k(n−k)

k+n−kk(n− k), k ≤ ⌊kk−1⌋.



◮ Remark. If ⌈kt(n)⌉ ≤ k ≤ ⌊kt−1(n)⌋ and (n− t)(k − t) iseven, the lower bound is attained on graphs Gk,n−t whichhave nk = n− t, nn−t = t, xk,n−t = t(n− t),

xk,k =(n−t)(k−t)

2and all other xij = 0.



◮ Remark. If ⌈kt(n)⌉ ≤ k ≤ ⌊kt−1(n)⌋ and (n− t)(k − t) iseven, the lower bound is attained on graphs Gk,n−t whichhave nk = n− t, nn−t = t, xk,n−t = t(n− t),

xk,k =(n−t)(k−t)

2and all other xij = 0.

◮ Extremal graph Gk,n−t is complete join G1 +G2 of graphsG1 and G2. G1 is regular graph on n− t vertices withdegree k − t and G2 is graph on t isolated vertices (withdegree 0). The complete join of two graphs is their graphunion with all the edges that connect the vertices of thefirst graph with the vertices of the second graph.


A quadratic programming model of the problem

min GA(G) =∑

k≤i≤n−1

i≤j≤n−1

2√ij

i+ jxi,j

◮

2xk,k + xk,k+1 + · · · + xk,n−1 = knk,

xk,k+1 + 2xk+1,k+1 + · · · + xk+1,n−1 = (k + 1)nk+1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,n−1 + xk+1,n−1 + · · · + 2xn−1,n−1 = (n− 1)nn−1,


◮ A quadratic programming model of the problem

min GA(G) =∑

k≤i≤n−1

i≤j≤n−1

2√ij

i+ jxi,j

◮

2xk,k + xk,k+1 + · · · + xk,n−1 = knk,

xk,k+1 + 2xk+1,k+1 + · · · + xk+1,n−1 = (k + 1)nk+1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,n−1 + xk+1,n−1 + · · · + 2xn−1,n−1 = (n− 1)nn−1,

nk + nk+1 + · · · + nn−1 = n,



min GA(G) =∑

k≤i≤n−1

i≤j≤n−1

2√ij

i+ jxi,j

◮

2xk,k + xk,k+1 + · · · + xk,n−1 = knk,

xk,k+1 + 2xk+1,k+1 + · · · + xk+1,n−1 = (k + 1)nk+1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,n−1 + xk+1,n−1 + · · · + 2xn−1,n−1 = (n− 1)nn−1,

nk + nk+1 + · · · + nn−1 = n,◮

xi,j ≤ ninj, k ≤ i < j ≤ n− 1,



min GA(G) =∑

k≤i≤n−1

i≤j≤n−1

2√ij

i+ jxi,j

◮

2xk,k + xk,k+1 + · · · + xk,n−1 = knk,

xk,k+1 + 2xk+1,k+1 + · · · + xk+1,n−1 = (k + 1)nk+1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,n−1 + xk+1,n−1 + · · · + 2xn−1,n−1 = (n− 1)nn−1,

nk + nk+1 + · · · + nn−1 = n,◮

xi,j ≤ ninj, k ≤ i < j ≤ n− 1,◮

xi,i ≤(

ni

2

)

, k ≤ i ≤ n− 1,



min GA(G) =∑

k≤i≤n−1

i≤j≤n−1

2√ij

i+ jxi,j

◮

2xk,k + xk,k+1 + · · · + xk,n−1 = knk,

xk,k+1 + 2xk+1,k+1 + · · · + xk+1,n−1 = (k + 1)nk+1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,n−1 + xk+1,n−1 + · · · + 2xn−1,n−1 = (n− 1)nn−1,

nk + nk+1 + · · · + nn−1 = n,◮

xi,j ≤ ninj, k ≤ i < j ≤ n− 1,◮

xi,i ≤(

ni

2

)

, k ≤ i ≤ n− 1,◮

xi,j , ni are non-negative integers, fork ≤ i ≤ j ≤ n− 1.


◮ The main results

◮ Theorem 1. If k ≥ ⌈k0⌉, where k0 = q0(n− 1), q0 ≈ 0.088is the unique positive root of equationq√q + q + 3

√q − 1 = 0 and if G ∈ G(k, n), then

GA(G) ≥ kn

2.

If k or n are even, this value is attained by regular graphsof degree k.


Extremal graph fork ≥ ⌈k0⌉

◮

��

��

��

��

��

``````

``````

``````

``````

```((((

((((((((

hhhhhhhh

hhhh

hhhhhhhhhhhh

((((((((((((

v v v v v v

v v v v v v

Fig. 1. Shape of extremal graph fork = 4.Conference 2016 – p. 14/35

Proof

◮ We will consider the problem of linear programming

min GA(G) =∑

k≤i≤n−1

i≤j≤n−1

2√ij

i+ jxi,j

subject to

2xk,k + xk,k+1 + · · · + xk,n−1 = knk,

xk,k+1 + 2xk+1,k+1 + · · · + xk+1,n−1 = (k + 1)nk+1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,n−1 + xk+1,n−1 + · · · + 2xn−1,n−1 = (n− 1)nn−1,


Proof


min GA(G) =∑

k≤i≤n−1

i≤j≤n−1

2√ij

i+ jxi,j

subject to

2xk,k + xk,k+1 + · · · + xk,n−1 = knk,

xk,k+1 + 2xk+1,k+1 + · · · + xk+1,n−1 = (k + 1)nk+1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,n−1 + xk+1,n−1 + · · · + 2xn−1,n−1 = (n− 1)nn−1,

nk + nk+1 + · · · + nn−1 = n,


Proof


min GA(G) =∑

k≤i≤n−1

i≤j≤n−1

2√ij

i+ jxi,j

subject to

2xk,k + xk,k+1 + · · · + xk,n−1 = knk,

xk,k+1 + 2xk+1,k+1 + · · · + xk+1,n−1 = (k + 1)nk+1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,n−1 + xk+1,n−1 + · · · + 2xn−1,n−1 = (n− 1)nn−1,

nk + nk+1 + · · · + nn−1 = n,◮

◮

xi,j ≥ 0, k ≤ i ≤ j ≤ n−1, ni ≥ 0, k ≤ i ≤ n−1.


Proof

◮ The basic variables areni, k ≤ i ≤ n− 1 andxk,k.

ni =xk,i + · · · + 2xi,i + · · · + xi,n−1

i, k + 1 ≤ i ≤ n− 1.


◮ Proof


ni =xk,i + · · · + 2xi,i + · · · + xi,n−1

i, k + 1 ≤ i ≤ n− 1.

nk = n−n−1∑

i=k+1

1

ixk,i −

∑

k+1≤i≤j≤n−1

(

1

i+

1

j

)

xi,j.


◮ Proof


ni =xk,i + · · · + 2xi,i + · · · + xi,n−1

i, k + 1 ≤ i ≤ n− 1.

nk = n−n−1∑

i=k+1

1

ixk,i −

∑

k+1≤i≤j≤n−1

(

1

i+

1

j

)

xi,j.

◮

xk,k =kn

2− 1

2

n−1∑

i=k+1

(

1 +k

i

)

xk,i −1

2

∑

k+1≤i≤j≤n−1

(

k

i+

k

j

)

xi,j.


◮ Proof

◮ Then

GA(G) =kn

2+

n−1∑

i=k+1

(

2√ki

k + i− k

2

(

1

k+

1

i

)

)

xk,i

+∑

k+1≤i≤j≤n−1

(

2√ij

i+ j− k

2

(

1

i+

1

j

))

xi,j.


Proof

◮ Since allai,j ≥ 0 for k ≤ i ≤ j ≤ n− 1, we conclude thatgeometric-arithmetic index will attains its minimum valuekn

2if

we putxi,j = 0 for all k ≤ i ≤ j ≤ n− 1, except forxk,k.Thus, we have proved

GA(G) ≥ kn

2.

Geometric-arithmetic index attains minimum valuekn2

if k or n

are even, on graphs forxk,k =kn2

, nk = n and all otherxi,j = 0andni = 0.


Casenk = n− 1

◮ Theorem 2. If nk = n− 1 and k ≤ ⌊k0⌋, wherek0 = q0(n− 1), q0 ≈ 0.0874 is the unique positive root ofequation q3 + 5q2 + 11q − 1 = 0, then

GA ≥ (n− 1)(k − 1)

2+

2(n− 1)√

k(n− 1)

k + n− 1= f1.

If (n− 1)(k − 1) is even, the lower bound attains on graphGk,n−1 which has nn−1 = 1, xk,n−1 = n− 1,

xk,k =(n−1)(k−1)

2and all others xij = 0.


Extremal graph fornk = n− 1

◮

w��

HHHH

HH

JJJJJJJ

JJJJJJJ

��

PPPPPP

t

t

t

t

tt

tt

t

t

��

PPPP

��

JJJJPP

PP

AAAA

��

��

��

DDDDDDDDDD

llllll,,,,,,

��

CCCCCCCC

QQQQ

QQ

Fig. 2. Shape of extremal graph fork = 5.


Proof of Casenk = n− 1

◮ In this casenk+1 + · · · + nn−1 = 1, which implies that existsk + 1 ≤ j ≤ n− 1, such thatnj = 1. If nk = n− 1, then

xk,k ≥ nk(nk−n+k)2

= (n−1)(k−1)2

. Putxk,k =(n−1)(k−1)

2+ yk,k.

We have2xk,k + xk,j = knk,

xk,j + 2xj,j = jnj.



◮ In this casenk+1 + · · · + nn−1 = 1, which implies that existsk + 1 ≤ j ≤ n− 1, such thatnj = 1. If nk = n− 1, then

xk,k ≥ nk(nk−n+k)2

= (n−1)(k−1)2

. Putxk,k =(n−1)(k−1)

2+ yk,k.

We have2xk,k + xk,j = knk,

xk,j + 2xj,j = jnj.

◮ After substitution ofxk,k, and sincexj,j = 0, nj = 1, we get

2yk,k + xk,j = (n− 1),

xk,j = j.

We haveyk,k =n−1−j

2andxk,k =

(n−1)(k−1)2

+ n−1−j

2.



◮ Geometric-arithmetic index is:

GA =(n− 1)(k − 1)

2+

n− 1− j

2+

2j√kj

k + j



◮ Geometric-arithmetic index is:

GA =(n− 1)(k − 1)

2+

n− 1− j

2+

2j√kj

k + j

◮ Since∂2GA∂j2

≤ 0, GA(j) is concave function forj ≥ k andattains its minimum value forj = n− 1 or j = k,

GA(n− 1) =(n− 1)(k − 1)

2+

2(n− 1)√

k(n− 1)

k + n− 1= f1,

GA(k) =nk

2= f0.



◮

f1 − f0 =−(n− 1 + k)2 + 4(n− 1)

√

k(n− 1)

2(k + n− 1).



◮

f1 − f0 =−(n− 1 + k)2 + 4(n− 1)

√

k(n− 1)

2(k + n− 1).

◮ f1 − f0 ≤ 0 if k ≤ ⌊k0⌋, wherek0 = q0(n− 1), q0 ≈ 0.0874 isthe unique positive root of equationq3 + 5q2 + 11q − 1 = 0,that is of q

√q + q + 3

√q − 1 = 0.


Casenk = n− 2

◮ Theorem 3. If nk = n− 2 and k ≤ ⌊k1⌋, where k1, is theunique positive root of equation f2 − f1 = 0, then

GA ≥ (n− 2)(k − 2)

2+2(n− 2)

√

k(n− 1)

k + n− 22(n−2) = f2.

If (n− 2)(k − 2) is even, the lower bound attains on graphGk,n−2 which has

nn−2 = 2, xk,n−2 = 2(n− 2), xk,k =(n−2)(k−2)

2and all

others xij = 0.


Extremal graph fornk = n− 2

◮

w w��

��

��

HHHHHHHHHHHH

��

��

��

XXXXXXXXXXXXXXXX

��

��

��

PPPPPPPPPPPPPPP

@@@@

��

TTTTT

��

%%%%%%%%

eeeeeeee

eeeeeeee

%%%%%%%%

HHHH

HHHH

HHHH

��

@@@@

��

��

TTTTT

XXXXXX

XXXXXX

XXXX

��

PPPP

PPPP

PPPP

PPP

��

t

t

t

t

t

t

t

t

t

t

t

t

t

t

�

PPP

��

``````

@

��

``````

@

�

PPP

Fig. 3. Shape of extremal graph fork = 4Conference 2016 – p. 25/35

Sketch of the proof of Theorem 3

◮ We consider three cases:a) nn−1 = 2, b) nn−1 = 1 andc) nn−1 = 0.



◮ We consider three cases:a) nn−1 = 2, b) nn−1 = 1 andc) nn−1 = 0.

◮ 2a. In this case we havexk,k =

(n−2)(k−2)2

, xk,n−1 = 2(n− 2), xn−1,n−1 = 1. We get

GA2 =(n− 2)(k − 2)

2+

2√

k(n− 1)

k + n− 12(n− 2) + 1.



◮ 2b. In this case there isk + 1 ≤ j ≤ n− 2, such thatnj = 1.

Thenxk,k =(n−2)(k−2)

2+ yk,k, xk,n−1 = nknn−1 =

n− 2, xj,j = 0, xj,n−1 = njnn−1 = 1.



◮ 2b. In this case there isk + 1 ≤ j ≤ n− 2, such thatnj = 1.

Thenxk,k =(n−2)(k−2)

2+ yk,k, xk,n−1 = nknn−1 =

n− 2, xj,j = 0, xj,n−1 = njnn−1 = 1.

◮ Similarly as in the casenk = n− 1, GA attains minimum valueGA1(k) for j = k or GA1(n− 2) for j = n− 2.

GA1(k) =(n− 1)(k − 1)

2+

2(n− 1)√

k(n− 1)

k + n− 1= f1,

GA1(n− 2) =(n− 2)(k − 2)

2+

1

2+

2√

k(n− 2)

k + n− 2(n− 3)

+2√

k(n− 1)

k + n− 1(n− 2) +

2√

(n− 2)(n− 1)

n− 2 + n− 1.



◮ 2c. We solve the next problem of linear programming

min∑

k≤i≤j≤n−2

2√ij

i+ jxij

2yk,k + xk,k+1 + · · · + xk,n−2 = 2(n− 2),

xk,k+1 + 2xk+1,k+1 + · · · + xk+1,n−2 = (k + 1)nk+1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,i + xk+1,i + · · · + xi,n−2 = ini,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xk,n−2 + xk+1,n−2 + · · · + 2xn−2,n−2 = (n− 2)nn−2,

nk+1 + nk+2 + · · · + nn−2 = 2



◮ We get

GA ≥ (n− 2)(k − 2)

2+

2√

k(n− 2)

k + n− 22(n− 2) = f2.



◮ We get

GA ≥ (n− 2)(k − 2)

2+

2√

k(n− 2)

k + n− 22(n− 2) = f2.

◮ Sincef2 ≤ GA2, f2 ≤ f1 andf2 ≤ GA1(n− 2), we get thatf2is minimum value of GA index in this case.


References

1 M. Aouchiche, P. Hansen,On a conjecture about the Randicindex, Discrete Mathematics, 307 (2), 2007, 262-265.


References


2 M. Aouchiche, P. Hansen, M. Zheng,Variable NeighborhoodSearch for Extremal Graphs. 18. Conjectures and Resultsabout the Randic Index, MATCH - Communications inMathematical and in Computer Chemistry, 56 (2006), 541-550.


References



3 M. Aouchiche, P. Hansen, M. Zheng,Variable NeighborhoodSearch for Extremal Graphs. 19. Further Conjectures andResults about the Randic Index, MATCH - Communicationsin Mathematical and in Computer Chemistry, 58 (2007),83-102.


References



3 M. Aouchiche, P. Hansen, M. Zheng,Variable NeighborhoodSearch for Extremal Graphs. 19. Further Conjectures andResults about the Randic Index, MATCH - Communicationsin Mathematical and in Computer Chemistry, 58 (2007),83-102.

4 G. Caporossi, I. Gutman, P. Hansen,Variable neighborhoodsearch for extremal graphs. 4. Chemical trees with extremalconnectivity index, Computers& Chemistry 23 (1999), 469-477. Conference 2016 – p. 30/35

References

5 K. Das, I. Gutman, B. Furtula,Survey on Geometric -Arithmetic Indices of Graphs, MATCH-Communications inMathematical and in Computer Chemistry, 65 (2011), 595-644.


References


6 T. Divnic, M. Milivojevi c, Lj. Pavlovic,Extremal graphs forthe geometric-arithmetic index with given minimum degree,Discrete Applied Mathematics, Vol. 162, 2014, 386-390.


References



7 T. Divnic, Lj. Pavlovic,Proof of the first part of theconjecture of Aouchiche and Hansen about the Randicindex, Discrete Applied Mathematics, Vol.161, Issues7-8,(May 2013), pp. 953-960.


References



7 T. Divnic, Lj. Pavlovic,Proof of the first part of theconjecture of Aouchiche and Hansen about the Randicindex, Discrete Applied Mathematics, Vol.161, Issues7-8,(May 2013), pp. 953-960.

8 Z. Du, B. ZHou, N. Trinajsitc,On Geometric-ArithmeticIndices of (Molecular) Trees, Unicyclic Graphs andBicyclic Graphs, MATCH-Communications in Mathematicaland in Computer Chemistry, 66 (2011), 681-697.


References

9 G. Fath-Tabar, B. Furtula, I. Gutman,A newgeometric-arithmetic index, Journal of MathematicalChemistry, Vol. 47, Issue 1 (2010), 477-486.


References


10 P. Hansen, D. Vukicevic,Variable Neighborhood Search forExtremal Graphs. 23. On the Randic index and thechromatic number, Discrete Mathematics,Vol. 309 (2009),Issue 13, pp. 4228-4234.


References



11 B. Liu, Lj. Pavlovic, J. Liu, T. Divnic, M. Stojanovic,On theconjecture of Aouchiche and Hansen about the Randicindex, Discrete Mathematics, Vol. 313, Issue 3, (2013), pp.225-235.


References



11 B. Liu, Lj. Pavlovic, J. Liu, T. Divnic, M. Stojanovic,On theconjecture of Aouchiche and Hansen about the Randicindex, Discrete Mathematics, Vol. 313, Issue 3, (2013), pp.225-235.

12 M. Mogharrab, G. Fath-Tabar,Some Bounds on GA1 Indexof Graphs, MATCH-Communications in Mathematical and inComputer Chemistry, 65 (2011), 33-38.


References

13 Lj. Pavlovic, T. Divnic,A quadratic programming approachto the Randic index, European Journal of OperationalResearch, 2007, Vol. 176, Issue 1, pp. 435-444.


References


14 Lj. Pavlovic, I. Gutman,Graphs with Extremal connectivityindex, Novi Sad J. Math, 2001, Vol. 31, No. 2, 53-58.


References



15 M. Randic,On characterization of molecular branching, J.Am. Chem. Soc., 97 (1975), 6609 -6615.


References



15 M. Randic,On characterization of molecular branching, J.Am. Chem. Soc., 97 (1975), 6609 -6615.

16 I. Tomescu, R. Marinescu-Ghemeci, G. Mihai,On densegraphs having minimum Randic index, Romanian J. ofInformation Science and Technology (ROMJIST), Vol. 12, (4),2009, pp. 455-465.


References



References


18 Y. Yuan, B. Zhou, N. Trinajstic,On geometric-arithmeticindex, Journal of Mathematical Chemistry, Vol. 47, Issue 2(2010), 833-841.


References


18 Y. Yuan, B. Zhou, N. Trinajstic,On geometric-arithmeticindex, Journal of Mathematical Chemistry, Vol. 47, Issue 2(2010), 833-841.

19 B. Zhou, I. Gutman, B. Furtula, Z. Du,On two types ofgeometric-arithmetic index, Chemical Physics Letters 482(2009), 153-155.


Conference in honour of prof. Dragoš Cvetkovic 2016

THANK YOU

FOR YOUR

ATTENTION!


conjecture for the geometric-arithmetic index with given minimum

Documents