4‐regular graphs without cut‐vertices having the same path layer matrix

Abstract

The path layer matrix of a graph G contains quantitative information about all possible paths in G. The entry (i; j) of this matrix is the number of paths in G having initial vertex i and length j. It is known that there are 4-regular graphs on 44 vertices having the same path layer matrix [177][178][179][180][181][182]. In this article, a pair of 4-regular graphs without cut-vertices on 18 vertices having the same path layer matrix are constructed, improving the upper bound for the least order of 4-regular graphs having the same path layer matrix from 44 to 18 and the upper bound for the least order of graphs without cut-vertices having the same path layer matrix from 31 to 18. ß

Figures of Graph Partitioning by Counting, Sequence and Layer Matrices

“…A layer matrix, a term initially used for tables expressing stratification by age in biological populations [56], was introduced into graph theory by Andrey A. Dobrynin (see [57][58][59]). Most of the studies involving the use of the layer matrix to differentiate between topological isomers were lead by Dobrynin [60,61] and Mircea V. Diudea [62], but other researchers also found uses for the layer matrices in their studies (see [63,64]). Haruro Hosoya were the first to introduce a counting polynomial (Z-counting polynomial in [65]; see general review on counting polynomials in chemistry in [66]) to characterize a graph, and George Pólya were the first to introduce the counting polynomial into graph theory to count the topological isomers [67].…”