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

Abstract

Let $\hat{I}“=(K_n,H)$ be a signed complete graph whose negative edges induce a subgraph $H$. Let $A(\hat{I}“)$ be the adjacency matrix of the signed graph $\hat{I}“$. The largest eigenvalue of $A(\hat{I}“)$ is called the index of $\hat{I}“$. 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 $\hat{I}“=(K_n,B)$ has the maximum index is determined.