Symmetric graphs - spectra and eigenvectors

2nd revised Edition.Davidson (1981) developed a general procedure, based on group representation theory, for determining the spectra of graphs distinguished by a certain rotational symmetry, with application to molecular graphs. In this paper a more general method, applicable to any arbitrarily arc weighted directed graph that has a non-trivial automorphism, and yielding both eigenvalues and eigenvectors, is developed. The proofs, elementary and straightforward, avoid the use of the theory of group characters altogether

Symmetric graphs - spectra and eigenvectors

Davidson (1981) developed a general procedure, based on group representation theory, for determining the spectra of graphs distinguished by a certain rotational symmetry, with application to molecular graphs. In this paper a more general method, applicable to any arbitrarily arc weighted directed graph that has a non-trivial automorphism, and yielding both eigenvalues and eigenvectors, is developed. The proofs, elementary and straightforward, avoid the use of the theory of group characters altogether

(2,6)-cages and their spectra

A (k, 6)-cage (k = 2, 3, 4, 5) is a 2-connected cubic plane graph that has only k-gons and hexagons as its faces. Continuing their work on (3, 6)-cages (2009) the authors investigate the combinatorial (topological and algebraic) structure of (2, 6)-cages and explicitly determine their eigenvalues and eigenvectors

Spectra of toroidal graphs

An n-fold periodic locally finite graph in the euclidean n-space may be considered the parent of an infinite class of n-dimensional toroidal finite graphs. An elementary method is developed which allows the characteristic polynomials of these graphs to be factored, in a uniform manner, into smaller polynomials, all of the same size. Applied to the hexagonal tessellation of the plane (the graphite sheet), this method enables the spectra and corresponding orthonormal eigenvector systems for all toroidal fullerenes and (3, 6) cages to be explicitly calculated. In particular, a conjecture of P.W. Fowler on the spectra of (3, 6) cages is proved

The asymptotic covering density of generalized Petersen graphs

Remark on the paper "Minimum vertex covers in the generalized Petersen graphs P(n; 2)" by M. Behzad, P. Hatami, and E.S. MahmoodianDedicated to Tomaz Pisanski on the occasion of his 60th birthdayThe covering density of a graph G=(V,E) is delta(G)= beta(G)/|V|where beta(G), the covering number, is the minimum number of vertices that represent all edges of G. The asymptotic covering density of the generalized Petersen graph is determined

O maksimalnom sparivanju i svojstvenim vrijednostima benzenoidnih grafova

In August 2003 the computer program GRAFFITI made conjecture 1001 stating that for any benzenoid graph, the size of a maximum matching equals the number of positive eigenvalues. Later, the authors learned that this conjecture was already known in 1982 to I. Gutman (Kragujevac). Here we present a proof of this conjecture and of a related theorem. The results are of some relevance in the theory of (unsaturated) polycyclic hydrocarbons.U kolovozu 2003. uporabom kompjutorskoga programa GRAFFITI naslućeno je da je za bilo koji benzenoidni graf maksimalno sparivanje jednako broju pozitivnih svojstvenih vrijednosti. Kasnije su autori saznali da je taj rezultat bio poznat već 1982. Ivanu Gutmanu (Kragujevac). U članku je dan rigorozan dokaz toga rezultata i odgovarajući teorem. Taj je rezultat od određene važnosti u teoriji policikličkih ugljikovodika

On Maximum Matchings and Eigenvalues of Benzenoid Graphs

In August 2003 the computer program GRAFFITI made conjecture 1001 stating that for any benzenoid graph, the size of a maximum matching equals the number of positive eigenvalues. Later, the authors learned that this conjecture was already known in 1982 to I. Gutman (Kragujevac). Here we present a proof of this conjecture and of a related theorem. The results are of some relevance in the theory of (unsaturated) polycyclic hydrocarbons

A Theorem for Counting Spanning Trees in General Chemical Graphs and Its Particular Application to Toroidal Fullerenes

A theorem is stated that enables the number of spanning trees in any finite connected graph to be calculated from two determinants that are easily obtainable from its cycles → edges incidence-matrix. The 1983 theorem of Gutman, Mallion and Essam (GME), applicable only to planar graphs, arises as a special case of what we are calling the Cycle Theorem (CT). The determinants encountered in CT are the same size as those arising in GME when planar graphs are under consideration, but CT is applicable to non-planar graphs as well. CT thus extends the conceptual and computational advantages of GME to graphs of any genus. This is especially of value as toroidal polyhexes and other carbon-atom species embedded on the torus, as well as on other non-planar surfaces, are presently of increasing interest. The Cycle Theorem is applied to certain classic, and other, graphs – planar and non-planar – including a typical toroidal polyhex