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. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The maximum cardinality of sextet patterns of Fn is the Clar number of Fn. It was shown that the Clar n...

A fullerene graph is a cubic bridgeless plane graph with all faces of size 5 and 6. We show that every fullerene graph on n vertices can be made bipartite by deleting at most √ 12n/5 edges, and has an independent set with at least n/2 − √ 3n/5 vertices. Both bounds are sharp, and we characterise the extremal graphs. This proves conjectures of Došlić and Vukičević, and of Daugherty. We deduce tw...

A fullerene graph F is a cubic 3-connected plane graph with exact 12 pentagons and other hexagons. Let M be a perfect matching of F . A cycle C of F is M -alternating if and only if the edges of C appear alternately in and off M . A set H of disjoint hexagons of F is called a resonant pattern if F has a perfect matching M such that all hexagons in H are M -alternating. A fullerene graph F is k-...

A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. In this paper we compute the first and the second geometric – arithmetic indices of Hamiltonian graphs. Then we apply our results to obtain some bounds for fullerene.

the eigenvalues of a graph is the root of its characteristic polynomial. a fullerene f is a 3-connected graphs with entirely 12 pentagonal faces and n/2 -10 hexagonal faces, where n isthe number of vertices of f. in this paper we investigate the eigenvalues of a class of fullerenegraphs.

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...

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...

An efficient numerical method for computing permanental polynomials of graphs is proposed. It adapts multi-entry expansion of FFT, and is parallel in nature. It is applied to fullerene-type graphs, and works for C56, while the largest fullerene computed before is C40. Extensive numerical computations show that the algorithm is fast and stable. © 2006 Elsevier B.V. All rights reserved. PACS: 02....

the wiener polarity index wp(g) of a molecular graph g of order n is the number ofunordered pairs of vertices u, v of g such that the distance d(u,v) between u and v is 3. in anearlier paper, some extremal properties of this graph invariant in the class of catacondensedhexagonal systems and fullerene graphs were investigated. in this paper, some new bounds forthis graph invariant are presented....

A fullerene graph is a 3 regular planar simple finite graph with pentagon or hexagon faces. In these graphs the number of pentagon faces is 12. Therefore, any fullerene graph can be characterized by number of its hexagon faces. In this note, for any h > 1, we will construct a fullerene graph with h hexagon faces. Then, using the leapfrogging process we will construct stable fullerenes with 20 +...