The Spectral Properties of $p$Sombor (Laplacian) Matrix of Graphs 
Received:May 20, 2022 Revised:November 24, 2022 
Key Words:
$p$Sombor matrix $p$Sombor Laplacian matrix $p$Sombor spectrum

Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11971180; 12271337) and the Characteristic Innovation Project of General Colleges and Universities in Guangdong Province (Grant No.2022KTSCX225). 

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Abstract: 
The Sombor index, which was recently introduced into chemical graph theory, can predict physicochemical 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 pSombor Laplacian eigenvalues. At last, we determine the Sombor spectrum of some special graphs. As a byproduct, we determine the spectral properties of Sombor matrix $(p=2)$, Zagreb matrix $(p=1)$ and inverse sum index matrix $(p=1)$. 
Citation: 
DOI:10.3770/j.issn:20952651.2023.03.003 
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