On the maximum spectral radius of connected graphs with a prescribed order and size

TL;DR

The paper resolves the problem of maximizing the spectral radius $\rho(G)$ over connected graphs with order $n$ and size $n-1+e$ by proving that extremality occurs within threshold graphs and is captured by the templates $\mathcal{D}_{n,e}$ and $\mathcal{V}_{n,e}$. A computer-assisted framework based on $\mathcal{T}$-subgraphs reduces the search to a finite set of templates, enabling complete verification for $e\le 130$, together with explicit comparison criteria via a cubic root threshold $\psi_e$ and a Bell-type argument for large $e$ that establishes $\mathcal{V}_{n,e}$ as the unique maximizer when $n\ge e+2+13\sqrt{e}$. The results yield a precise, regime-based extremal characterization and provide computable mechanisms (via polynomials $\mathcal{P}_{\mathcal{T}_1(G)}$, $\mathcal{P}_{\mathcal{T}(G)}$, and $\mathcal{Q}_{G_1,G_2}$) to decide which graph attains the maximum spectral radius. Overall, the work advances extremal spectral graph theory by delivering complete solutions for $e\le 130$ and sharp large-$n$ bounds, incorporating both combinatorial and algebraic methods alongside computer-assisted verification.

Abstract

The spectral radius of a graph is the largest modulus of an eigenvalue of its adjacency matrix. Let $\mathcal{C}_{n, e}$ be the set of all the connected simple graphs with $n$ vertices and $n - 1 + e$ edges. Here, we solve the spectral radius maximization problem on $\mathcal{C}_{n, e}$ when $e \le 130$ or $n \ge e + 2 + 13\sqrt{e}$.

Theorems & Definitions (43)

• Theorem 1.1: bruhoff1985
• Theorem 1.2: bruhoff1985
• Theorem 1.3: rowlinson1988
• Theorem 1.4: brusol1986
• Theorem 1.5: brusol1986
• Theorem 1.6: brusol1986
• Theorem 1.7: cvetrow1988
• Theorem 1.8: bell1991
• Theorem 1.9: oleroydri2002
• Proposition 1.10