Cubic graphs without cut‐vertices having the same path layer matrix

Abstract

The path layer matrix (or path degree sequence) 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 cubic graphs on 62 vertices having the same path layer matrix (A. A. Dobrynin. J Graph Theory 17 (1993) 1±4). A new upper bound of 36 vertices for the least order of such cubic graphs is established. This bound is realized by cubic graphs without cut-vertices. ß

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

“…Since G 18 and H 18 are a pair of nonisomorphic 4-regular graphs without cutvertices having the same path layer matrix, we have the following. Dobrynin [6] verified that an equality of the path layer matrices for 3-regular graphs of order p 18 is a sufficient condition for their isomorphism. We verify that an equality of the path layer matrices for 4-regular graphs of order p 15 is a sufficient condition for their isomorphism.…”

“…Investigations of these matrices deal with finding a pair of nonisomorphic graphs having some specified properties such that both graphs have the same matrix. Among these properties we mention the girth, cyclomatic number, and planarity of graphs [1][2][3][4][5][6][7][8][9][10][12][13][14][15][16][17][18][19].…”

“…By constructing three pairs of nonisomorphic r-regular graphs having the same path layer matrices, Yang Yuangsheng [19] and co-workers proved that pð4Þ 44; pð5Þ 48; pð6Þ 51: Let f 2 be the least order for pairs of nonisomorphic graphs without cut-vertices having the same path layer matrix. By constructing a pair of nonisomorphic 3regular graphs without cut-vertices having the same path layer matrix for p ¼ 36, and a pair of nonisomorphic graphs without cut-vertices having the same path layer matrix for p ¼ 31, Dobrynin [6] proved that pð3Þ 36; f 2 31:…”