A graph G is singular of nullity (> 0), if its adjacency matrix A is singular, with the eigenvalue zero of multiplicity . A singular graph having a 0-eigenvector, x, with no zero entries, is called a core graph.We place particular emphasis on nut graphs, namely the core graphs of nullity one. Through symmetry considerations of the automorphism group of the graph, we study relations among the en...

Using elements of the structural theory of matchings and a recently proved conjecture concerning bricks, it is shown that every n-extendable brick (except K4, C6 and the Petersen graph) with p vertices and q edges contains at least q − p + (n − 1)!! perfect matchings. If the girth of such an n-extendable brick is at least five, then this graph has at least q − p + nn−1 perfect matchings. As a c...

A fullerene, which is a 3-connected cubic plane graph whose faces are pentagons and hexagons, is cyclically 5 edge-connected. For a fullerene of order n, it is customary to index the eigenvalues in non-increasing order λ1 ≥ λ2 ≥ · · · ≥ λn. It is known that the largest eigenvalue is 3. Let R = {fi | i ∈ Zl} be a set of l faces of a fullerene F such that fi is adjacent to fi+1, i ∈ Zl, via an ed...

By a fullerene, we mean a trivalent plane graph Γ = (V,E, F ) with only hexagonal and pentagonal faces. It follows easily from Euler’s Formula that each fullerene has exactly 12 pentagonal faces. The simplest fullerene is the graph of the dodecahedron with 12 pentagonal faces and no hexagonal faces. It is frequently easier to work with the duals to the fullerenes: geodesic domes, i.e. triangula...

This paper is dedicated to propose an algorithm in order to generate the certain isomers of some well-known fullerene bases. Furthermore, we enlist the bipartite edge frustration correlated with some of symmetrically distinct innite families of fullerenes generated by the oered process.

In this study, the effect of fullerene electron mobility on the composition of paraphenylenediamine and stability was studied. Using quantum chemistry calculations, the first combination of paraffenylenediamine in a single-full-time region connected with fullerene through carbon atoms was reported. Experimental research was simulated and optimized using GIS software. Then the NBO calculations w...

We consider the following question, motivated by the enumeration of fullerenes. A fullerene patch is a 2-connected plane graph G in which inner faces have length 5 or 6, non-boundary vertices have degree 3, and boundary vertices have degree 2 or 3. The degree sequence along the boundary is called the boundary code of G. We show that the question whether a given sequence S is a boundary code of ...

Background & Objective: The potent antioxidant property of fullerene C60 nanoparticles and their derivatives has been demonstrated in a wide range of in vitro and in vivo studies. Hence, we examined the effects of fullerene C60 on the oxidative stress parameters at brain and liver of rats in normal situation. Materials & Methods: The study was performed in two groups of Wistar rats (each group...

The smallest number of edges that have to be deleted from a graph G to obtain a bipartite spanning subgraph is called the bipartite edge frustration of G and denoted by φ(G). In this paper our recent results on computing this quantity for hierarchical product of graphs are reported. We also present a fast algorithm for computing edge frustration index of (3, 6)−fullerene graphs.