Multiplicity Bounds for Arbitrary Eigenvalues of Connected Signed Graphs

The study of eigenvalue multiplicities plays a central role in the spectral theory of signed graphs, extending several classical results from the unsigned setting. While most existing work focuses on the nullity of a signed graph (the multiplicity of the eigenvalue $0$), much less is known for arbitrary eigenvalues. In this paper, we establish a sharp upper bound for the multiplicity $m(G_\sigma, \lambda)$ of any real eigenvalue $\lambda$ of a connected signed graph $G_\sigma$ in terms of its girth. Our main result shows that $$m(G_\sigma, \lambda) \le n - g(G_\sigma) + 2,$$ where $n$ is the number of vertices and $g(G_\sigma)$ is the girth. We prove that equality holds if and only if $G_\sigma$ is switching equivalent to one of the following extremal families: \begin{itemize} \item[(i)] a balanced complete graph with $\lambda = -1$; \item[(ii)] an antibalanced complete graph with $\lambda = 1$; or \item[(iii)] a balanced complete bipartite graph with $\lambda = 0$. \end{itemize} This fully extends and generalizes the known result for the nullity case ($\lambda = 0$), originally due to Wu et al.\ (2022), to the entire eigenvalue spectrum. Our approach combines Cauchy interlacing, switching equivalence, and a structural analysis of induced cycles in signed graphs. We also provide a characterization of eigenvalues with multiplicity $1$ and $2$ for signed cycles, and include examples illustrating the sharpness and spectral behavior of the extremal families.