The Spectral Properties of $p$-Sombor (Laplacian) Matrix of Graphs

DOI：10.3770/j.issn:2095-2651.2023.03.003

 作者 单位 刘合超 华南师范大学数学科学学院, 广东 广州 510631 尤利华 华南师范大学数学科学学院, 广东 广州 510631

最近在化学图论引入的Sombor指数可以预测分子的物理化学性质. 本文从代数的角度来研究($p$-)Sombor指数的性质. $p$-Sombor矩阵$\mathcal{S}_{p}(G)$是一个$n$阶方阵, 当$v_{i}\sim v_{j}$时, 其$(i,j)$位置的元素为$((d_{i})^{p}+(d_{j})^{p})^{\frac{1}{p}}$, 否则为$0$, 其中$d_{i}$表示图$G$中顶点$v_{i}$的度. 该矩阵推广了著名的Zagreb矩阵$(p=1)$、Sombor矩阵$(p=2)$和inverse sum indeg矩阵$(p=-1)$. 本文找到了一对$p$-Sombor非同谱的等能量图, 并确定了$p$-Sombor(拉普拉斯)谱半径的一些界. 然后刻画了具有$k$个不同$p$-Sombor拉普拉斯特征值的连通图的性质. 最后确定了一些特殊图的Sombor谱. 作为推论, 确定了Sombor矩阵$(p=2)$, Zagreb矩阵$(p=1)$和inverse sum indeg矩阵$(p=-1)$的谱性质.

The Sombor index, which was recently introduced into chemical graph theory, can predict physico-chemical properties of molecules. In this paper, we investigate the properties of ($p$-)Sombor index from an algebraic viewpoint. The $p$-Sombor matrix $\mathcal{S}_{p}(G)$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to $((d_{i})^{p}+(d_{j})^{p})^{\frac{1}{p}}$ if $v_{i}\sim v_{j}$, and 0 otherwise, where $d_{i}$ denotes the degree of vertex $v_{i}$ in $G$. The matrix generalizes the famous Zagreb matrix $(p=1)$, Sombor matrix $(p=2)$ and inverse sum index matrix $(p=-1)$. In this paper, we find a pair of $p$-Sombor noncospectral equienergetic graphs and determine some bounds for the $p$-Sombor (Laplacian) spectral radius. Then we describe the properties of connected graphs with $k$ distinct p-Sombor Laplacian eigenvalues. At last, we determine the Sombor spectrum of some special graphs. As a by-product, we determine the spectral properties of Sombor matrix $(p=2)$, Zagreb matrix $(p=1)$ and inverse sum index matrix $(p=-1)$.