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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

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

Figure: Methane CH4.

Alkanes

Figure: Ethane C2H6.

Alkanes

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

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 .

(v , a)

(v , b)

(v , c)

(v , d)

(a, v)

(b, v)

(c , v)

(d , v)

Figure: The stellation G∗ of methane CH4.

Figure: The stellation G∗ of ethane C2H6.

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

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

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

The Main Lemma

Lemma

The Main Lemma

Lemma

The Main Lemma

Lemma

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.

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.

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.

[email protected]

Thank You!

[email protected]

Thank You!

[email protected]

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