The Normalized Laplacian Spectrum of Subdivision Vertex-Edge Corona for Graphs
Received:June 12, 2018  Revised:October 10, 2018
Key Word: normalized Laplacian spectrum   cospectral graphs   spanning trees   subdivision vertex-edge corona  
Fund ProjectL:Supported by the Young Scholars Science Foundation of Lanzhou Jiaotong University (Grant Nos.2016014; 2017004; 2017021), the Education Foundation of Gansu Province (Grant No.2017A-021) and the National Natural Science Foundation of China (Grant Nos.11461038; 61163010).
Author NameAffiliation
Muchun LI Institute of Applied Mathematics, Lanzhou Jiaotong University, Gansu 730070, P. R. China 
You ZHANG Institute of Applied Mathematics, Lanzhou Jiaotong University, Gansu 730070, P. R. China 
Fei WEN Institute of Applied Mathematics, Lanzhou Jiaotong University, Gansu 730070, P. R. China 
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Abstract:
      A 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 $i$-th vertex in $V(G_{1})$ to each vertex in the $i$-th copy of $G_{2}$ and $i$-th vertex of $I(G_1)$ to each vertex in the $i$-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's constant and the number of the spanning trees of $G_1^S\circ (G_2^V\cup G_3^E)$ on three regular graphs.
Citation:
DOI:10.3770/j.issn:2095-2651.2019.03.001
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