两类四圈图的奇异性

doi:10.13371/j.cnki.chin.q.j.m.2023.04.002

数学季刊 ›› 2023, Vol. 38 ›› Issue (4): 349-359.doi: 10.13371/j.cnki.chin.q.j.m.2023.04.002

两类四圈图的奇异性

1. School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China; 2. School of Mathematics and Statistics, Qinghai Normal University, Xining 810008, China

收稿日期:2022-11-14 出版日期:2023-12-30 发布日期:2023-12-30
通讯作者: MA Hai-cheng (1965-), male, native of Menyuan, Qinghai, professor of Qinghai Minzu University, engages in algebra graph theory; E-mail: qhmymhc@163.com
作者简介: YOU Xiao-jie (1998-), female, native of Taiyuan, Shanxi, graduate student of Qinghai Minzu University, engages in algebra; MA Hai-cheng (1965-), male, native of Menyuan, Qinghai, professor of Qinghai Minzu University, engages in algebra graph theory; Zhang Bin (1992-), male, native of Xining, Qinghai, lecturer of Qinghai Minzu University, engages in computational biology; LI Ya-lan (1986-), female, native of Xining, Qinghai, associate professor of Qinghai Normal University, engages in algebra graph theory.
基金资助:
Supported by National Natural Science Foundation of China (Grant No.11561056); National Natural Science Foundation of Qinghai Provence (Grant No. 2022-ZJ-924); Innovation Project of Qinghai Minzu University (Grant No.07M2022002).

Singularity of Two Kinds of Four Cycle Graphs

1. School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China; 2. School of Mathematics and Statistics, Qinghai Normal University, Xining 810008, China

Received:2022-11-14 Online:2023-12-30 Published:2023-12-30
Contact: MA Hai-cheng (1965-), male, native of Menyuan, Qinghai, professor of Qinghai Minzu University, engages in algebra graph theory; E-mail: qhmymhc@163.com
About author: YOU Xiao-jie (1998-), female, native of Taiyuan, Shanxi, graduate student of Qinghai Minzu University, engages in algebra; MA Hai-cheng (1965-), male, native of Menyuan, Qinghai, professor of Qinghai Minzu University, engages in algebra graph theory; Zhang Bin (1992-), male, native of Xining, Qinghai, lecturer of Qinghai Minzu University, engages in computational biology; LI Ya-lan (1986-), female, native of Xining, Qinghai, associate professor of Qinghai Normal University, engages in algebra graph theory.
Supported by:
Supported by National Natural Science Foundation of China (Grant No.11561056); National Natural Science Foundation of Qinghai Provence (Grant No. 2022-ZJ-924); Innovation Project of Qinghai Minzu University (Grant No.07M2022002).

摘要/Abstract

摘要： Let G be a finite simple graph and A(G) be its adjacency matrix. Then G is singular if A(G) is singular. The graph obtained by bonding the starting vertices and ending vertices of three paths P_a1, P_a2, P_a3is called θ-graph, represented by θ(a1,a2,a3). The graph obtained by bonding the two end vertices of the path P_s to the vertices of the θ(a₁,a₂,a₃) and θ(b₁,b₂,b₃) of degree three, respectively, is denoted by α(a₁,a₂,a₃,s,b₁,b₂,b₃) and called α-graph. β-graph is denoted when β(a₁,a₂,a₃,b₁,b₂,b₃) =α(a₁,a₂,a₃,1,b₁,b₂,b₃). In this paper, we give the necessary and sufficient conditions for the singularity of α-graph and β-graph, and prove that the probability that a random given α-graph and β-graph is a singular graph is equal to 1423/2048 and 733/1024 , respectively.

关键词: Adjacency matrix, Singular graph, Nullity, Probability

Abstract: Let G be a finite simple graph and A(G) be its adjacency matrix. Then G is singular if A(G) is singular. The graph obtained by bonding the starting vertices and ending vertices of three paths P_a1, P_a2, P_a3is called θ-graph, represented by θ(a1,a2,a3). The graph obtained by bonding the two end vertices of the path P_s to the vertices of the θ(a₁,a₂,a₃) and θ(b₁,b₂,b₃) of degree three, respectively, is denoted by α(a₁,a₂,a₃,s,b₁,b₂,b₃) and called α-graph. β-graph is denoted when β(a₁,a₂,a₃,b₁,b₂,b₃) =α(a₁,a₂,a₃,1,b₁,b₂,b₃). In this paper, we give the necessary and sufficient conditions for the singularity of α-graph and β-graph, and prove that the probability that a random given α-graph and β-graph is a singular graph is equal to 1423/2048 and 733/1024 , respectively.

Key words: Adjacency matrix, Singular graph, Nullity, Probability

中图分类号:

O157.5

攸晓杰, 马海成, 张斌, 李雅兰. 两类四圈图的奇异性[J]. 数学季刊, 2023, 38(4): 349-359.

YOU Xiao-jie, MA Hai-cheng, ZHANG Bin, LI Ya-lan. Singularity of Two Kinds of Four Cycle Graphs[J]. Chinese Quarterly Journal of Mathematics, 2023, 38(4): 349-359.

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两类四圈图的奇异性

Singularity of Two Kinds of Four Cycle Graphs

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