On Reduced Reciprocal Randi´c Energy of Lexocographic product of graphs

Abstract

Let G = G1[G2] represent the lexicographic product of two graphs with m+ n vertices and mn
edges. The vertex set of G is given byV (G) = {wij = (uij , vij ) : 1 ≤ i ≤ m, 1 ≤ j ≤ n}. As introduced in [4],
the reduced reciprocal Randi´c matrix of a graph G with n vertices, denoted by RRR(G), is an n × n matrix
whose (i, j)th entry is defined as q (dvi − 1)(dvj − 1) if the vertices vi and vj are adjacent, and 0 otherwise. The
reduced reciprocal Randi´c energy of a graph, denoted by RRRE(G), is defined as the sum of the absolute values
of the eigenvalues of RRR(G). In this work, we investigate the reduced reciprocal Randi´c energy RRRE(G)
for several lexicographic product graphs, including RRR(Km[Kn]), RRR(Km[Cn])and RRR(Cm[Cn]).