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