Signed bicyclic graphs with minimal index

Communications in Combinatorics and Optimization

article

Maurizio Brunetti¹  Adriana Ciampella¹ 

[1] Dipartimento di Matematica e Applicazioni R. Caccioppoli. Universita' di Napoli Federico II

关键词: Signed graph;    Bicyclic Graph;    Index;    Extremal Graph Theory;   

DOI  :  10.22049/cco.2022.27346.1241

学科分类：社会科学、人文和艺术（综合）

来源: Azarbaijan Shahide Madani Universit

【 摘 要 】

The index $\lambda_1(\Gamma)$ of a signed graph $\Gamma=(G,\sigma)$ is just the largest eigenvalue of its adjacency matrix. For any $n \geqslant 4$ we identify the signed graphs achieving the minimum index in the class of signed bicyclic graphs with $n$ vertices. Apart from the $n=4$ case, such graphs are obtained by considering a starlike tree with four branches of suitable length (i.e.\ four distinct paths joined at their end vertex $u$) with two additional negative independent edges pairwise joining the four vertices adjacent to $u$. As a by-product, all signed bicyclic graphs containing  a theta-graph and whose index is less than $2$ are detected.

【 授权许可】

CC BY-SA