The normalized Laplacian spectrum of subdivision vertex-edge corona for graphs
Received:July 12, 2018  Revised:November 12, 2018
Key Word: normalized Laplacian spectrum, cospectral graphs, spanning tree, subdivision vertex-edge corona
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 Author Name Affiliation E-mail You Zhang Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China zhangyoumath@163.com Muchun LI Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China limuchunmath@163.com Fei Wen Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China wenfei@mail.lzjtu.cn
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A \emph{subdivision vertex-edge corona} $G_1^S\circ (G_2^V\cup G_3^E)$ is a graph that consists of $S(G_1)$, $|V(G_1)|$ copies of $G_2$ and $|I(G_1)|$ copies of $G_3$ by joining the \emph{i}-th vertex in $V(G_{1})$ to each vertex in the \emph{i}-th copy of $G_{2}$ and \emph{i}-th vertex of $I(G_1)$ to each vertex in the \emph{i}-th copy of $G_3$. In this paper, we determine the normalized Laplacian spectrum of $G_1^S\circ (G_2^V\cup G_3^E)$ in terms of the corresponding normalized Laplacian spectra of three connected regular graphs $G_{1}$, $G_{2}$ and $G_{3}$. As applications, we construct some non-regular normalized Laplacian cospectral graphs. In addition, we also give the multiplicative degree-Kirchhoff index, the Kemeny&amp;#39;&amp;#39;s constant and the number of the spanning tree of $G_1^S\circ (G_2^V\cup G_3^E)$ on three regular graphs.