Fullerene graphs are mathematical models of fullerene molecules. The Wiener (r,s)-complexity a graph G with vertex set V(G) is the number pairwise distinct values (r,s)-transmiss...

Fullerene graphs are 3-connected 3-regular planar graphs with only pentagonal and hexagonal faces. We show that the diameter of a fullerene graph G of order n is at least √ 24n−15−3 6 and at most n/5 + 1. Moreover, if G is not a (5, 0)-nanotube its diameter is at most n/6+5/2. As a consequence, we improve the upper bound on the saturation number of fullerene graphs. We also report an improved l...

ABSTRACT. Suppose G is a graph, A(G) its adjacency matrix and f(G, x)=x^n+a_(n-1)x^(n-1)+... is the characteristic polynomial of G. The matching polynomial of G is defined as M(G, x) = x^n-m(G,1)x^(n-2) + ... where m(G,k) is the number of k-matchings in G. In this paper, we determine the relationship between 2k-th coefficient of characteristic polynomial, a_(2k), and k-th coefficient of matchin...

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the...

We show that every pair of hexagons in a fullerene graph satisfying the isolated pentagon rule (IPR) forms a resonant pattern. This solves a problem raised by Ye et al. [SIAM J. Discrete Math. 23(2):2009, p. 1023–1044].

The stable marriage problem (SM) and its many variants are extensively-studied combinatorial problems with many practical applications. The n-dimensional stable marriage model is proposed, which is initially motivated by a problem arising from chemical graph theory, and an algorithm is given accordingly. This model solves the zero clustering problem of chemical graph polynomials successfully, w...

(Received May 2, 2012) Graph theoretic fullerenes are designed to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. A totally symmetric Kekulé structure in a fullerene is a set of independent edges which is fixed by all symmetries of the fullerene. It was suggested in a paper by S.J.Austin, J.Baker, P.W.Fowler, D.E.Manolopoulos and in a p...

Fullerene graphs are trivalent plane graphs with only hexagonal and pentagonal faces. They are often used to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. A totally symmetric Kekulé structure in a fullerene graph is a set of independent edges which is fixed by all symmetries of the fullerene and molecules with totally symmetric Kekulé...

A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. By the Euler formula one can see that Fn has 12 pentagonal and n/2 – 10 hexagonal faces. Let G = (V, E) be a connected graph with the vertices set V = V(G) and the edges set E = E(G), without loops and multiple edges. The distance d(x,y) betwe...

The independence number of the graph of a fullerene, the size of the largest set of vertices such that no two are adjacent (corresponding to the largest set of atoms of the molecule, no pair of which are bonded), appears to be a useful selector in identifying stable fullerene isomers. The experimentally characterized isomers with 60, 70 and 76 atoms uniquely minimize this number among the class...