The Normalized Laplacian Spectrum of Subdivision Vertex-Edge Corona for Graphs

DOI：10.3770/j.issn:2095-2651.2019.03.001

 作者 单位 李沐春 兰州交通大学应用数学研究所, 甘肃 兰州 730070 张友 兰州交通大学应用数学研究所, 甘肃 兰州 730070 文飞 兰州交通大学应用数学研究所, 甘肃 兰州 730070

剖分点-边冠$G_1^S\circ (G_2^V\cup G_3^E)$是由$S(G_1)$, $|V(G_1)|$个$G_2$和$|I(G_1)|$个$G_3$通过将$V(G_{1})$中的每一个点连接到拷贝在该点上$G_{2}$ 中的每一个点, 以及$I(G_1)$中的每一个点连接到拷贝在该点上$G_{3}$中的每一个点所构成的图. 首先根据三个连通正则图$G_{1}$, $G_{2}$和$G_{3}$的规范拉普拉斯特征根确定了图$G_1^S\circ (G_2^V\cup G_3^E)$的规范拉普拉斯谱, 其次构造了一类非正则$\mathcal{L}$-同谱图. 此外，我们还给出了图$G_1^S\circ (G_2^V\cup G_3^E)$的多重度-Kirchhoff 指标、Kemeny常数和生成树数目.

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.