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Eigenvalues of Saturated Hydrocarbons
Craig Larson(joint work with Doug Klein)
Virginia Commonwealth UniversityRichmond, VA
CanaDAMJune 12, 2013
Doug’s Idea
Establish a simple model for saturated hydrocarbons that
I captures what every chemist “knows”—that alkane MOeigenvalues are half positive and half negative,
I suggesting that further mathematical results for this class areachievable,
I and using chemical graph theory to describe the electronicstructure of molecules other than conjugated hydrocarbons.
Doug’s Idea
Establish a simple model for saturated hydrocarbons that
I captures what every chemist “knows”—that alkane MOeigenvalues are half positive and half negative,
I suggesting that further mathematical results for this class areachievable,
I and using chemical graph theory to describe the electronicstructure of molecules other than conjugated hydrocarbons.
Doug’s Idea
Establish a simple model for saturated hydrocarbons that
I captures what every chemist “knows”—that alkane MOeigenvalues are half positive and half negative,
I suggesting that further mathematical results for this class areachievable,
I and using chemical graph theory to describe the electronicstructure of molecules other than conjugated hydrocarbons.
Doug’s Idea
Establish a simple model for saturated hydrocarbons that
I captures what every chemist “knows”—that alkane MOeigenvalues are half positive and half negative,
I suggesting that further mathematical results for this class areachievable,
I and using chemical graph theory to describe the electronicstructure of molecules other than conjugated hydrocarbons.
Saturated Hydrocarbons
DefinitionA saturated hydrocarbon is a connected graph whose vertices haveboth degrees one and four and no other degrees.
Figure: Cyclobutane C4H8.
Saturated Hydrocarbons
DefinitionA saturated hydrocarbon is a connected graph whose vertices haveboth degrees one and four and no other degrees.
Figure: Cyclobutane C4H8.
Alkanes
DefinitionAn alkane is an acyclic saturated hydrocarbon.
Figure: Methane CH4.
Alkanes
DefinitionAn alkane is an acyclic saturated hydrocarbon.
Figure: Methane CH4.
Alkanes
DefinitionAn alkane is an acyclic saturated hydrocarbon.
Figure: Ethane C2H6.
Alkanes
DefinitionAn alkane is an acyclic saturated hydrocarbon.
Figure: Ethane C2H6.
n Connected graphs with ∆ ≤ 4 Saturated Hydrocarbons
5 21 16 78 07 353 18 1,929 59 12,207 12
10 89,402 4411 739,335 19012 6,800,637 99513 68,531,618 6,21114 748,592,936 45,116
Table: All counts are for non-isomorphic graphs.
Figure: The unique saturated hydrocarbon with 7 atoms.
Molecular Orbitals
Molecular Orbitals
The Stellation Model
DefinitionThe stellation of a graph G is the graph G ∗
I with vertices V (G ∗) = ∪ab∈E(G){(a, b), (b, a)}.I Vertices (x , y), (z ,w) ∈ V (G ∗) are adjacent if, and only if,
either x = z or both x = w and y = z .
I Then E ∗ext = {(a, b)(b, a) : a ∼ b in G},I E ∗int = {(a, b)(a, c) : a ∼ b and a ∼ c in G}, and
I E (G ∗) = E ∗int ∪ E ∗ext .
The Stellation Model
(v , a)
(v , b)
(v , c)
(v , d)
(a, v)
(b, v)
(c , v)
(d , v)
Figure: The stellation G∗ of methane CH4.
The Stellation Model
Figure: The stellation G∗ of ethane C2H6.
The Stellation Model
Figure: The stellation G∗ of cyclobutane C4H8.
Some Precursers
From the Chemical Literature:
I C. Sandorfy, LCAO MO calculations on saturatedhydrocarbons and their substituted derivatives, CanadianJournal of Chemistry 33 (1955), no. 8, 1337–1351.
I K. Fukui, H. Kato, and T. Yonezawa, Frontier electron densityin saturated hydrocarbons, Bulletin of the Chemical Society ofJapan 34 (1961), no. 3, 442–445.
I J. A. Pople and D. P. Santry, A molecular orbital theory ofhydrocarbons, Molecular Physics 7 (1964), no. 3, 269–286.
From the Mathematical Literature:
I Schmidt & Haynes, 1990, Dunbar & Haynes, 1996, Favaron,&c.
I T. Shirai, The spectrum of infinite regular line graphs,Transactions of the American Mathematical Society 352(2000), no. 1, 115–132.
Some Precursers
From the Chemical Literature:
I C. Sandorfy, LCAO MO calculations on saturatedhydrocarbons and their substituted derivatives, CanadianJournal of Chemistry 33 (1955), no. 8, 1337–1351.
I K. Fukui, H. Kato, and T. Yonezawa, Frontier electron densityin saturated hydrocarbons, Bulletin of the Chemical Society ofJapan 34 (1961), no. 3, 442–445.
I J. A. Pople and D. P. Santry, A molecular orbital theory ofhydrocarbons, Molecular Physics 7 (1964), no. 3, 269–286.
From the Mathematical Literature:
I Schmidt & Haynes, 1990, Dunbar & Haynes, 1996, Favaron,&c.
I T. Shirai, The spectrum of infinite regular line graphs,Transactions of the American Mathematical Society 352(2000), no. 1, 115–132.
Some Precursers
From the Chemical Literature:
I C. Sandorfy, LCAO MO calculations on saturatedhydrocarbons and their substituted derivatives, CanadianJournal of Chemistry 33 (1955), no. 8, 1337–1351.
I K. Fukui, H. Kato, and T. Yonezawa, Frontier electron densityin saturated hydrocarbons, Bulletin of the Chemical Society ofJapan 34 (1961), no. 3, 442–445.
I J. A. Pople and D. P. Santry, A molecular orbital theory ofhydrocarbons, Molecular Physics 7 (1964), no. 3, 269–286.
From the Mathematical Literature:
I Schmidt & Haynes, 1990, Dunbar & Haynes, 1996, Favaron,&c.
I T. Shirai, The spectrum of infinite regular line graphs,Transactions of the American Mathematical Society 352(2000), no. 1, 115–132.
A Property of Stellated Graphs
The external edges form a perfect matching.
A Property of Stellated Graphs
The external edges form a perfect matching.
Weights
For a stellated graph G ∗ with vertex set V (G ∗) = {v1, . . . , vn} wedefine a weighted adjacency matrix Aw as follows:
I Awi ,j = 1 if vivj is an external edge in G ∗,
I Awi ,j = w if vivj is a internal edge, and
I Awi ,j = 0 otherwise.
Aw is the weighted adjacency matrix for G ∗.
Weights
For a stellated graph G ∗ with vertex set V (G ∗) = {v1, . . . , vn} wedefine a weighted adjacency matrix Aw as follows:
I Awi ,j = 1 if vivj is an external edge in G ∗,
I Awi ,j = w if vivj is a internal edge, and
I Awi ,j = 0 otherwise.
Aw is the weighted adjacency matrix for G ∗.
Weights
For a stellated graph G ∗ with vertex set V (G ∗) = {v1, . . . , vn} wedefine a weighted adjacency matrix Aw as follows:
I Awi ,j = 1 if vivj is an external edge in G ∗,
I Awi ,j = w if vivj is a internal edge, and
I Awi ,j = 0 otherwise.
Aw is the weighted adjacency matrix for G ∗.
Weights
For a stellated graph G ∗ with vertex set V (G ∗) = {v1, . . . , vn} wedefine a weighted adjacency matrix Aw as follows:
I Awi ,j = 1 if vivj is an external edge in G ∗,
I Awi ,j = w if vivj is a internal edge, and
I Awi ,j = 0 otherwise.
Aw is the weighted adjacency matrix for G ∗.
Weights
For a stellated graph G ∗ with vertex set V (G ∗) = {v1, . . . , vn} wedefine a weighted adjacency matrix Aw as follows:
I Awi ,j = 1 if vivj is an external edge in G ∗,
I Awi ,j = w if vivj is a internal edge, and
I Awi ,j = 0 otherwise.
Aw is the weighted adjacency matrix for G ∗.
The Determinant
DefinitionThe determinant of an n × n square matrix A with entries Ai ,j is
det A =∑σ∈Sn
sgn(σ)n∏
i=1
Ai ,σ(i),
where Sn is the set of permutations from [n] to itself and sgn(σ) is1 if σ can be written as an even number of permutations and −1otherwise.
The Main Lemma
Lemma
I Let G be a graph with a perfect matching M,
I with edges in M having unit weight,
I and remaining edges weighted w in a interval I ⊆ Rcontaining 0,
I and corresponding weighted adjacency matrix Aw .
If det Aw 6= 0 for all w ∈ I then Aw has half positive and halfnegative eigenvalues for each w ∈ I .
The Main Lemma
Lemma
I Let G be a graph with a perfect matching M,
I with edges in M having unit weight,
I and remaining edges weighted w in a interval I ⊆ Rcontaining 0,
I and corresponding weighted adjacency matrix Aw .
If det Aw 6= 0 for all w ∈ I then Aw has half positive and halfnegative eigenvalues for each w ∈ I .
The Main Lemma
Lemma
I Let G be a graph with a perfect matching M,
I with edges in M having unit weight,
I and remaining edges weighted w in a interval I ⊆ Rcontaining 0,
I and corresponding weighted adjacency matrix Aw .
If det Aw 6= 0 for all w ∈ I then Aw has half positive and halfnegative eigenvalues for each w ∈ I .
The Main Lemma
Lemma
I Let G be a graph with a perfect matching M,
I with edges in M having unit weight,
I and remaining edges weighted w in a interval I ⊆ Rcontaining 0,
I and corresponding weighted adjacency matrix Aw .
If det Aw 6= 0 for all w ∈ I then Aw has half positive and halfnegative eigenvalues for each w ∈ I .
The Main Lemma
Lemma
I Let G be a graph with a perfect matching M,
I with edges in M having unit weight,
I and remaining edges weighted w in a interval I ⊆ Rcontaining 0,
I and corresponding weighted adjacency matrix Aw .
If det Aw 6= 0 for all w ∈ I then Aw has half positive and halfnegative eigenvalues for each w ∈ I .
Alkane Eigenvalues
TheoremIf G is an alkane then its stellation G ∗ has half positive and halfnegative eigenvalues for any real number internal edge weight w.
Alkane Eigenvalues
TheoremIf G is an alkane then its stellation G ∗ has half positive and halfnegative eigenvalues for any real number internal edge weight w.
Unicyclic Saturated Hydrocarbon Eigenvalues
LemmaIf C2k is an even cycle with edge weights alternating between 1and w ∈ (0, 1) then det C2k 6= 0.
Unicyclic Saturated Hydrocarbon Eigenvalues
LemmaIf C2k is an even cycle with edge weights alternating between 1and w ∈ (0, 1) then det C2k 6= 0.
Unicyclic Saturated Hydrocarbon Eigenvalues
LemmaIf G is a saturated hydrocarbon formed from a cycle with twopendants attached to each vertex then the stellated graph G ∗ withunit weight external edges and internal edges with weightw ∈ [0, 1) has half positive and half negative eigenvalues.
Unicyclic Saturated Hydrocarbon Eigenvalues
LemmaIf G is a saturated hydrocarbon formed from a cycle with twopendants attached to each vertex then the stellated graph G ∗ withunit weight external edges and internal edges with weightw ∈ [0, 1) has half positive and half negative eigenvalues.
Unicyclic Saturated Hydrocarbon Eigenvalues
TheoremIf G is a unicyclic saturated hydrocarbon then its stellation G ∗ hashalf positive and half negative eigenvalues for any internal edgeweight w ∈ [0, 1).
A General Theorem
TheoremAny stellated saturated hydrocarbon with external edges of unitweight and internal edges with weights w ∈ [0, c) has half positiveand half negative eigenvalues, for some molecule-dependentconstant c > 0.
A Conjecture
Conjecture
Any stellated saturated hydrocarbon with external edges of unitweight and internal edges with weights w ∈ [0, 1) has half positiveand half negative eigenvalues.
Thank You!
D. J. Klein and C. E. Larson,Eigenvalues of Saturated Hydrocarbons,
Journal of Mathematical Chemistry 51(6) 2013, 1608–1618.
Thank You!
D. J. Klein and C. E. Larson,Eigenvalues of Saturated Hydrocarbons,
Journal of Mathematical Chemistry 51(6) 2013, 1608–1618.
Thank You!
D. J. Klein and C. E. Larson,Eigenvalues of Saturated Hydrocarbons,
Journal of Mathematical Chemistry 51(6) 2013, 1608–1618.