The Index of Signed Graphs with Forbidden Subgraphs

Abstract

A signed graph $\Gamma$ is the graph whose edges get signs $\pm \, 1$. The index of $\Gamma$ is the largest eigenvalue of its adjacency matrix. For a family $\mathcal {F}$ of signed graphs, a signed graph $\Gamma$ is said to be $\mathcal {F}$-free if $\Gamma$ contains no member in $\mathcal {F}$ as its subgraph. The family consisting of all $\mathcal {F}$-free graphs on n vertices is denoted by $\mathbb {G}(n,\mathcal {F})$. If $\mathcal {F}=\{F\}$, we simply write $\mathcal {F}$ as F. Let $K^+_{n}$ and $C^+_{n}$ be the complete graph of order n and cycle of order n whose edges get signs $+\,1$, respectively. In this paper, we, respectively, characterize the extremal graphs possessing the maximum index among $\mathbb {G}(n,K_s^+)$ with $s\ge 2$, $\mathbb {G}(n,\mathcal {C})$ with $\mathcal {C}=\{C^+_l:3\le l\le n\}$ and $\mathbb {G}(n,\mathcal {C}_{2k})$ with $\mathcal {C}_{2k}=\{C^+_{2k}:2\le k\le \lfloor \frac{n}{2}\rfloor \}$.