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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.09768 |
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| _version_ | 1866917137167155200 |
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| author | Alfuraidan, Monther R. Khan, Suliman |
| author_facet | Alfuraidan, Monther R. Khan, Suliman |
| contents | 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 $m(G_σ, λ)$ of any real eigenvalue $λ$ of a connected signed graph $G_σ$ in terms of its girth. Our main result shows that \[ m(G_σ, λ) \le n - g(G_σ) + 2, \] where $n$ is the number of vertices and $g(G_σ)$ is the girth. We prove that equality holds if and only if $G_σ$ is switching equivalent to one of the following extremal families: \begin{itemize}
\item[(i)] a balanced complete graph with $λ= -1$;
\item[(ii)] an antibalanced complete graph with $λ= 1$; or
\item[(iii)] a balanced complete bipartite graph with $λ= 0$. \end{itemize} This fully extends and generalizes the known result for the nullity case ($λ= 0$), 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicity Bounds for Arbitrary Eigenvalues of Connected Signed Graphs Alfuraidan, Monther R. Khan, Suliman Combinatorics 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 $m(G_σ, λ)$ of any real eigenvalue $λ$ of a connected signed graph $G_σ$ in terms of its girth. Our main result shows that \[ m(G_σ, λ) \le n - g(G_σ) + 2, \] where $n$ is the number of vertices and $g(G_σ)$ is the girth. We prove that equality holds if and only if $G_σ$ is switching equivalent to one of the following extremal families: \begin{itemize} \item[(i)] a balanced complete graph with $λ= -1$; \item[(ii)] an antibalanced complete graph with $λ= 1$; or \item[(iii)] a balanced complete bipartite graph with $λ= 0$. \end{itemize} This fully extends and generalizes the known result for the nullity case ($λ= 0$), 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. |
| title | Multiplicity Bounds for Arbitrary Eigenvalues of Connected Signed Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.09768 |