A note on commutative graphs

doi:10.3785/j.issn.1008-9497.2024.02.005

Journal of Zhejiang University (Science Edition)

2024, Vol. 51

Issue (2): 172-177 DOI: 10.3785/j.issn.1008-9497.2024.02.005

Mathematics and Computer Science

A note on commutative graphs

Han WU^1,²,Fenjin LIU¹(

),Fanqi SHANG^1,³,Yanhong ZHOU¹,Haotong RUAN¹

^1.School of Science，Chang'an University，Xi'an 710064，China
^2.Department of Mathematics，Nanjing University，Nanjing 210093，China
^3.Qingdao Innovation and Development Base of Harbin Engineering University，Qingdao 266000，Shandong Province，China

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Abstract

Two simple graphs are commutative if there exists a labelling of their vertices such that their adjacency matrices can commute. This paper gives three necessary conditions ensuring the commutativity of certain graphs from Perron vectors, the number of main eigenvalues, the regularity of graphs. Then we construct new commutative graphs by graph Kronecker product, Cartesian product and circulant matrix. Finally, for two commutative graphs, we provide two algorithms that can express one adjacency matrix as the matrix polynomial of another adjacency matrix with distinct eigenvalues, and compare their merits. Commutative graphs sharing common eigenvectors are essential to the study of spectral graph theory.

Key words： commutative graph regular graph circulant graph Kronecker product Cartesian product

Received: 09 May 2022 Published: 08 March 2024

CLC:

O 157.5

Corresponding Authors: Fenjin LIU E-mail: fenjinliu@chd.edu.cn

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	Han WU
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Cite this article:

Han WU,Fenjin LIU,Fanqi SHANG,Yanhong ZHOU,Haotong RUAN. A note on commutative graphs. Journal of Zhejiang University (Science Edition), 2024, 51(2): 172-177.

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https://www.zjujournals.com/sci/EN/Y2024/V51/I2/172

可交换图的一些注记

如果存在一种顶点标号，使得2个简单图的邻接矩阵可交换，则称2个简单图可交换。首先，从图的Perron向量、主特征值数量、正则性三方面给出了可交换图的必要条件。然后，借助矩阵的克罗内克积、图的笛卡尔积及循环矩阵，构造了新的可交换图。最后，将一个邻接矩阵表示为另一个特征值互异的邻接矩阵的矩阵多项式，给出了2种算法，并比较了二者的优劣。可交换图存在公共的特征向量，对图谱理论研究具有重要意义。

关键词： 可交换图, 正则图, 循环图, 克罗内克积, 笛卡尔积

Fig.1 Commutative graphs with different number of main eigenvalues

Fig.2 Non-regular commutative graphs

G

and

G ¯

Fig.3 Connected graphs with 6 vertices and the number of main eigenvalues less than 3

图的序号	可交换图的序号
$G 6$	$G 6$
$G 7$	$G 7$ ， $G 13$
$G 8$	$G 8$
$G 9$	$G 9$ ， $G 17$ ， $G 19$ ， $G 20$
$G 10$	$G 10$ ， $G 16$
$G 11$	$G 11$
$G 12$	$G 12$
$G 13$	$G 7$ ， $G 13$
$G 14$	$G 14$
$G 15$	$G 15$
$G 16$	$G 10$ ， $G 16$
$G 17$	$G 9$ ， $G 17$ ， $G 19$ ， $G 20$
$G 18$	$G 18$
$G 19$	$G 9$ ， $G 17$ ， $G 19$
$G 20$	$G 9$ ， $G 17$ ， $G 20$
$G 21$	$G 21$
$G 22$	$G 22$
$G 23$	$G 23$

Table 1 Connected graphs with six vertices two main eigenvalues and their commutative graphs

Fig.4 A pair of commutative graphs that can be expressed as matrix polynomial of their adjacency matrices


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