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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.
clarson@vcu.edu
Thank You!
D. J. Klein and C. E. Larson,Eigenvalues of Saturated Hydrocarbons,
Journal of Mathematical Chemistry 51(6) 2013, 1608–1618.
clarson@vcu.edu
Thank You!
D. J. Klein and C. E. Larson,Eigenvalues of Saturated Hydrocarbons,
Journal of Mathematical Chemistry 51(6) 2013, 1608–1618.
clarson@vcu.edu