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| ON THE NORMALIZED LAPLACIAN SPECTRUM OF A NEW JOIN OF TWO GRAPHS |
| Xianzhang Wu,Lili Shen |
| (College of Math. and Computer Science, Fuzhou University, Fuzhou 350116, Fujian, PR China) |
| DOI： |
| 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 $G_{1}\star G_{2}$. |
| Key words: spectrum; $SSG$-vertex join; normalized Laplacian cospectral graphs; normalized Laplacian energy; degree Kirchhoff index |