On the maximal index of $C_4^-$-free unbalanced signed graphs of prescribed size

Abstract

The extremal problem of determining the graphs with maximum spectral radius among all graphs with a prescribed number of edges was resolved by Rowlinson [Linear Algebra Appl. 110 (1988) 43--53]. A signed graph $Γ=(G,σ)$ is called $C_k^-$-free if it contains no negative $k$-cycle as a signed subgraph. The index of a signed graph is defined as the largest eigenvalue of its adjacency matrix. This paper addresses the maximization of the index of unbalanced signed graphs. For signed graphs of a prescribed order, Wang and Lin [Discrete Appl. Math. 372 (2025) 164--172] determined the maximum index of $C_4^{-}$-free unbalanced signed graphs. Subsequently, Wang, Peng and Hou [Discrete Math. 349 (2026) 114871] characterized the unbalanced signed graphs of prescribed size with maximum index. This paper treats the edge spectral Turán-type problem for $C_4^-$-free unbalanced signed graphs. The all-positive signed graph with underlying graph $G$ is denoted by $(G,+)$. Evidently, $A((G,+))=A(G)$. In particular, $(H_{d,t},+)$ is the all-positive signed graph whose underlying graph is $H_{d,t}$, where $H_{d,0}=K_d$ and for $t>0$, $H_{d,t}$ is obtained from $K_d$ by adding a single vertex adjacent to exactly $t$ vertices of $K_d$. Let $m-3=\binom{d}{2}+t$ with $0\le t\le d-1$. Up to switching isomorphism and the addition of isolated vertices, the unique extremal unbalanced signed graph is obtained by attaching a negative triangle to a specified vertex of the all-positive signed graph $(H_{d,t},+)$.