Anti-Kekulé number of the {(3, 4), 4}-fullerene*

Abstract

A {(3,4),4}-fullerene graphG is a 4-regular plane graph with exactly eight triangular faces and other quadrangular faces. An edge subset S of G is called an anti-Kekulé set, if G − S is a connected subgraph without perfect matchings. The anti-Kekulé number of G is the smallest cardinality of anti-Kekulé sets and is denoted by akG. In this paper, we show that 4≤akG≤5; at the same time, we determine that the {(3, 4), 4}-fullerene graph with anti-Kekulé number 4 consists of two kinds of graphs: one of which is the graph H1 consisting of the tubular graph Qnn≥0, where Qn is composed of nn≥0 concentric layers of quadrangles, capped on each end by a cap formed by four triangles which share a common vertex (see Figure 2 for the graph Qn); and the other is the graph H2, which contains four diamonds D1, D2, D3, and D4, where each diamond Di1≤i≤4 consists of two adjacent triangles with a common edge ei1≤i≤4 such that four edges e1, e2, e3, and e4 form a matching (see Figure 7D for the four diamonds D1 − D4). As a consequence, we prove that if G∈H1, then akG=4; moreover, if G∈H2, we give the condition to judge that the anti-Kekulé number of graph G is 4 or 5.

Honeycomb Rhombic Torus Vertex-Edge Based Resolvability Parameters and Its Application in Robot Navigation

“…By referring to chemical graph theory as opposed to graph theory, it is highlighted that one is permitted to rely on intuitive knowledge of many ideas and theorems rather than on rigorous mathematical proofs. A molecule is turned into a molecular graph by transforming bonds into edges and vertices [1], [2]. The spatial configuration of atoms in a molecule is known as chemical structure.…”

The Structural Properties of (2,6)-Fullerenes

“…A (4, 6)-fullerene is the molecular graph of a boron-nitrogen fullerene. The structural properties, such as connectivity, extendability, resonance, anti-Kekulé number, are very useful for studying the number of perfect matchings in a graph [2,3]. And the number of perfect matchings is closely related to the stability of molecular graphs [4][5][6][7][8].…”

“…Therefore, many articles have studied the structural properties of graphs in both mathematics and chemistry [9][10][11]. Fullerene graphs are bicritical, cyclically 5-edge-connected, 2-extendable and 1-resonant [12][13][14][15]; Boron-nitrogen fullerene graphs are bipartite, 3-connected, 1-extendable, 2-resonant, and have the forcing number at least two [16,17]; A (3,6)-fullerene is 1-extendable, 1-resonant and has the connectivity 2 or 3 [18,19]. This paper is mainly concerned with the structural properties of (2, 6)-fullerenes.…”

The Structural Properties of (2, 6)-Fullerenes

“…A (4, 6)-fullerene is the molecular graph of a boron-nitrogen fullerene. The structural properties, such as connectivity, extendability, resonance, and anti-Kekulé number, are very useful for studying the number of perfect matchings in a graph [2,3]. And the number of perfect matchings is closely related to the stability of molecular graphs [4][5][6][7][8].…”

“…In 2011, Jiang et al proved that boron-nitrogen fullerene graphs have the forcing number at least two [17]. A (3,6)-fullerene is 1-extendable, and has the connectivity 2 or 3 [18]. In 2012, Yang et al showed that each hexagon of a (3,6)-fullerene with connectivity 2 is not resonant, and each hexagon of a (3,6)-fullerene with connectivity 3 is resonant except for one graph [19].…”