On Spectral Radius and Energy of a Graph with Self-Loops

4区工程技术 Q1 Mathematics Mathematical Problems in Engineering Pub Date : 2024-05-10 DOI:10.1155/2024/7056478

Deekshitha Vivek Anchan, Gowtham H. J., Sabitha D’Souza

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引用次数: 0

Abstract

The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron–Frobenius theory, it will be one among its eigenvalues. In this paper, Perron–Frobenius theory for adjacency matrix of graph with self-loops will be explored. Specifically, it discusses the nontrivial existence of Perron–Frobenius eigenvalue and eigenvector pair in the matrix , where denotes the number of self-loops. Also, Koolen–Moulton type bound for the energy of graph is explored. In addition, the existence of a graph with self-loops for every odd energy is proved.

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论带自循环图的谱半径和能量

方阵的谱半径是其特征值绝对值的最大值。假设一个正方形矩阵是非负矩阵，那么根据 Perron-Frobenius 理论，该矩阵的特征值将是其特征值中的一个。本文将探讨有自循环图的邻接矩阵的 Perron-Frobenius 理论。具体来说，本文将讨论矩阵中的 Perron-Frobenius 特征值和特征向量对的非偶数存在性，其中，Perron-Frobenius 表示自循环的数量。同时，还探讨了图的能量的 Koolen-Moulton 类型约束。此外，还证明了在每个奇数能量下都存在具有自循环的图。

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来源期刊

Mathematical Problems in Engineering 工程技术-工程：综合

CiteScore

4.00

自引率

0.00%

发文量

2853

审稿时长

4.2 months

期刊介绍： Mathematical Problems in Engineering is a broad-based journal which publishes articles of interest in all engineering disciplines. Mathematical Problems in Engineering publishes results of rigorous engineering research carried out using mathematical tools. Contributions containing formulations or results related to applications are also encouraged. The primary aim of Mathematical Problems in Engineering is rapid publication and dissemination of important mathematical work which has relevance to engineering. All areas of engineering are within the scope of the journal. In particular, aerospace engineering, bioengineering, chemical engineering, computer engineering, electrical engineering, industrial engineering and manufacturing systems, and mechanical engineering are of interest. Mathematical work of interest includes, but is not limited to, ordinary and partial differential equations, stochastic processes, calculus of variations, and nonlinear analysis.