| The Normalized Laplacian Spectrum of Subdivision Vertex-Edge Corona for Graphs |
| Received:June 12, 2018 Revised:October 10, 2018 |
| Key Words:
normalized Laplacian spectrum cospectral graphs spanning trees subdivision vertex-edge corona
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| Fund Project: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). |
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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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