The maximum index of signed complete graphs whose negative edges induce a bicyclic graph

Abstract

Let $Γ=(K_n,H)$ be a signed complete graph whose negative edges induce a subgraph $H$. Let $A(Γ)$ be the adjacency matrix of the signed graph $Γ$. The largest eigenvalue of $A(Γ)$ is called the index of $Γ$. In this paper, the index of all the signed complete graphs whose negative edges induce a bicyclic graph $B$ is investigated. Specifically, the structure of the bicyclic graph $B$ such that $Γ=(K_n,B)$ has the maximum index is determined.