图的剖分点-边冠运算下的规范拉普拉斯谱

A subdivision vertex-edge corona G_1~S?(∪ 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?(G_2~V∪ 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?(G_2~V∪ G_3~E) on three regular graphs.

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

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.