Extremal eigenvalues of graphs embedded on surfaces

TL;DR

The paper resolves the eigenvalue-extremal problem for graphs embeddable on a surface with Euler genus $\gamma$ by establishing tight spectral-radius bounds and a structural description of extremal graphs. It shows that extremal graphs arise from the base join $K_2 \nabla P_{n-2}$ with exactly $3\gamma$ additional edges, with explicit sharp bounds $\rho(G) \in \bigl(\rho_0+\frac{3\gamma-1}{n},\rho_0+\frac{3\gamma-0.95}{n}\bigr)$ where $\rho_0=\tfrac{3}{2}+\sqrt{2n-\frac{15}{4}}$, improving previous results. The authors also derive exact extremal graphs for projective-plane and torus embeddings, showing that for large $n$, $\rho(G)\leq \rho(K_2 \nabla K_4^{n-2})$ and $\rho(G)\leq \rho(K_2 \nabla K_5^{n-2})$ respectively, with characterizations of equality. They further analyze low-genus cases (e.g., $\gamma\in\{1,2\}$) to identify specific extremal graphs, leveraging a blend of spectral methods and detailed structural/embedding arguments. Overall, the work provides a novel framework for understanding eigenvalue-extremal problems on surface graphs and lays out open questions about complete characterizations for large $n$.

Abstract

Graph theory on surfaces extends classical graph structures to topological surfaces, providing a theoretical foundation for characterizing the embedding properties of complex networks in constrained spaces. The study of bounding the spectral radius $ρ(G)$ of graphs on surfaces has a rich history that dates back to the 1990s. In this paper, we establish tight bounds for graphs of order $n$ that are embeddable on a surface with Euler genus $γ$. Specifically, if graph $G$ achieves the maximum spectral radius, then \begin{equation*} \begin{array}{ll} \frac32\!+\!\sqrt{2n\!-\!\frac{15}4}\!+\!\frac{3γ\!-\!1}{n}<ρ(G)<\frac32\!+\!\sqrt{2n\!-\!\frac{15}4}\!+\!\frac{3γ\!-\!0.95}{n}, \end{array} \end{equation*} which improves upon the earlier bound $ρ(G)\leq2+\sqrt{2n+8γ-6}$ by Ellingham and Zha [JCTB, 2000]. Furthermore, we prove that any extremal graph is obtained from $K_2 \nabla P_{n-2}$ by adding exactly $3γ$ edges, where `$\nabla$' means the join product. As a corollary, for $γ= 0$ and $n \geq 4.5 \times 10^6$, the graph $K_2 \nabla P_{n-2}$ is the unique planar extremal graph, thereby confirming a long-standing conjecture resolved by Tait and Tobin [JCTB, 2017]. Let $K_r^n$ be the graph of order $n$ obtained by attaching two paths of nearly equal length to two distinct vertices of $K_r$. Integrating spectral techniques with considerable structural analysis on surface graphs, we further derive the following sharp bounds: $ρ(G) \leq ρ(K_2 \nabla K_4^{n-2})$ for projective-planar graphs, and $ρ(G) \leq ρ(K_2 \nabla K_5^{n-2})$ for toroidal graphs. Our study presents a novel framework for exploring the eigenvalue-extremal problem on surface graphs with high Euler genus.

Figures (9)

• Figure 1: Embeddings of graphs on torus, projective plane and sphere.
• Figure 2: The plane graphs $H_{i}$ and $H'_{i}$, where $i\geq 1$.
• Figure 3: The operations ($b$) and ($c$).
• Figure 4: An illustration of the local structures of $\widetilde{G"}$, $\widetilde{G'}$ and $\widetilde{G"'}$.
• Figure 5: Extending an $(\ell\!-\!2)$-walk to an $\ell$-walk by inserting a closed 2-walk.

Theorems & Definitions (69)

• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Definition 2.1