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arXiv math.CO bot @[email protected] · 2025-06-02 · 07:25 UTC

The stability of independence polynomials of complete bipartite graphs

Guo Chen, Bo Ning, Jianhua Tu
https://arxiv.org/abs/2505.24381 https://arxiv.org/pdf/2505.24381 https://arxiv.org/html/2505.24381

arXiv:2505.24381v1 Announce Type: new
Abstract: The independence polynomial of a graph is termed {\it stable} if all its roots are located in the left half-plane $\{z \in \mathbb{C} : \mathrm{Re}(z) \leq 0\}$, and the graph itself is also referred to as stable. Brown and Cameron (Electron. J. Combin. 25(1) (2018) \#P1.46) proved that the complete bipartite graph $K_{1,n}$ is stable and posed the question: \textbf{Are all complete bipartite graphs stable?}
We answer this question by establishing the following results:
\begin{itemize}
\item The complete bipartite graphs $K_{2,n}$ and $K_{3,n}$ are stable.
\item For any integer $k\geq0$, there exists an integer $N(k)\in \mathbb{N}$ such that $K_{m,m+k}$ is stable for all $m>N(k)$.
\item For any rational $\ell> 1$, there exists an integer $N(\ell) \in \mathbb{N}$ such that whenever $m >N(\ell)$ and $\ell \cdot m$ is an integer, $K_{m, \ell \cdot m}$ is \textbf{not} stable.
\end{itemize}

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arXiv math.CO bot @[email protected] · 2025-05-20 · 07:30 UTC

Generalized spectral characterization of signed bipartite graphs

Songlin Guo, Wei Wang, Lele Li
https://arxiv.org/abs/2505.12446 https://arxiv.org/pdf/2505.12446 https://arxiv.org/html/2505.12446

arXiv:2505.12446v1 Announce Type: new
Abstract: Let $\Sigma$ be an $n$-vertex controllable or almost controllable signed bipartite graph, and let $\Delta_\Sigma$ denote the discriminant of its characteristic polynomial $\chi(\Sigma; x)$. We prove that if (\rmnum{1}) the integer $2^{ -\lfloor n/2 \rfloor }\sqrt{\Delta _{\Sigma}}$ is squarefree, and (\rmnum{2}) the constant term (even $n$) or linear coefficient (odd $n$) of $\chi(\Sigma; x)$ is $\pm 1$, then $\Sigma$ is determined by its generalized spectrum. This result extends a recent theorem of Ji, Wang, and Zhang [Electron. J. Combin. 32 (2025), \#P2.18], which established a similar criterion for signed trees with irreducible characteristic polynomials.

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