On the Normalized Laplacian Spectrum of a New Join of Two Graphs

Abstract

Given graphs $G_1$ and $G_2,$ we define a graph operation on $G_1$ and $G_2$, namely the $SSG$-vertex join of $G_1$ and $G_2,$ denoted by $G_1 \star G_2.$ Let $S(G)$ be the subdivision graph of $G.$ The $SSG$-vertex join $G_1\star G_2$ is the graph obtained from $S(G_1)$ and $S(G_2)$ by joining each vertex of $G_1$ with each vertex of $G_2.$ In this paper, when $G_i (i = 1, 2)$ is a regular graph, we determine the normalized Laplacian spectrum of $G_1 \star G_2.$ As applications, we construct many pairs of normalized Laplacian cospectral graphs, the normalized Laplacian energy, and the degree Kirchhoff index of $G_1 \star G_2.$