On a conjecture of distance spectral extremal problems

Abstract: Brualdi and Hoffman proposed a well-known problem of determining the graph with maximum adjacency spectral radius among all graphs with given size $m$. Early work by Friedland and Stanley addressed some specific cases. This problem was later completely solved by Rowlinson and recently revisited by Cheng and Weng. Pioneering work on the distance matrix was carried out by Graham and Pollak, as well as by Graham and Lovász. The distance spectral radius $ρ(G)$ of a connected graph $G$ is the largest eigenvalue of its distance matrix. In this paper, we completely solve the problem of characterizing the connected graph with minimum distance spectral radius among all graphs with size $m$. Let $\mathcal{G}(m)$ be the class of connected graphs with $m$ edges. For every $m \ge 3$, let $n$ be the unique integer satisfying $\binom{n-1}{2} < m \le \binom{n}{2}$, and we write $m = \binom{n-1}{2} + s$ with $1 \le s \le n-1$. Recently, Lin and Zhou [Adv. in Appl. Math. 173 (2026)] investigated the graph in $\mathcal{G}(m)$ that minimizes $ρ(G)$ in the range $s \ge \max{ \frac{n-6}{2}, 1}$. However, the problem is much more difficult in the remaining range $1 \le s \le \frac{n-7}{2}$, and they conjectured that the unique minimizer is $\overline{P_{n,s+1}}$, the complement of a balanced disjoint union of paths. Using novel matrix analysis, we solve this conjecture in the affirmative. Moreover, we provide a new unified proof for the entire range $1 \le s \le n-1$. The key ingredients in our proof argument include an innovative comparison principle for the distance spectral radius, an increment analysis of $Φ$-functions on paths and cycles, an argument for balancing path lengths, and a walk enumeration technique via the Neumann series.

• The paper proves that the complement of balanced disjoint paths uniquely minimizes the distance spectral radius among connected graphs with prescribed edges.
• It introduces a novel matrix comparison framework and closed-form Φ-function expressions to analyze spectral increments and graph structure.
• The unified proof eliminates local switching arguments, providing a robust methodology for tackling broader extremal spectral problems.

Distance Spectral Extremal Problems: Resolution of the Lin–Zhou Conjecture

Introduction

The paper addresses the extremal problem of determining, among all connected graphs with a prescribed number of edges $m$, which graph minimizes the distance spectral radius $\rho(G)$, where $\rho(G)$ is the largest eigenvalue of the distance matrix $D(G)$. The distance spectral problems are rooted in foundational work by Graham, Pollak, and Lovász, as well as the adjacency spectral extremal tradition starting with Brualdi and Hoffman, and its resolution by Rowlinson. Unlike the adjacency spectral radius which is maximized by the complete graph, the distance spectral radius is minimized by the complete graph and maximized by the path, reflecting fundamentally opposite spectral behaviors with respect to connectivity and separation.

Previous work by Lin and Zhou—in the regime where $s \geq \max\{\frac{n-6}{2}, 1\}$—had identified the minimizer as the complement of a balanced disjoint union of paths, denoted $\overline{P_{n,s+1}}$. However, for smaller values of $s$, the problem proved resistant to the standard local edge-switching arguments because global monotonicity is violated and configurations with cycles could not be improved via simple local operations. Lin and Zhou conjectured that $\overline{P_{n,s+1}}$ remains the unique minimizer for all $1 \leq s \leq \frac{n-7}{2}$. This paper solves their conjecture in the affirmative and provides a unified proof for the full range $1 \leq s \leq n-1$, developing a global matrix-analytic framework.

Problem Formulation and Structural Lemmas

Let $\rho(G)$0 denote the class of connected graphs with $\rho(G)$1 edges. For each $\rho(G)$2, let $\rho(G)$3 be the unique integer such that $\rho(G)$4, and write $\rho(G)$5 with $\rho(G)$6. The central extremal graph is $\rho(G)$7, whose complement structure is a balanced disjoint union of $\rho(G)$8 paths. The structural lemma (Theorem [LZ2026]) establishes that minimizers must have complement consisting of disjoint cycles and paths, with exactly $\rho(G)$9 path components, restricting the problem to combinatorial configurations of these components.

Matrix-Analysis Comparison Principle

The fundamental technical innovation is a matrix comparison principle that eliminates the need for local switching arguments. By expressing $\rho(G)$0, where $\rho(G)$1, the paper derives that $\rho(G)$2 is characterized as the unique root $\rho(G)$3 of $\rho(G)$4. Defining the $\rho(G)$5-function for each component $\rho(G)$6 as $\rho(G)$7 and the global $\rho(G)$8-function as their sum, the comparison of spectral radii reduces to comparing $\rho(G)$9-functions for different configurations.

This approach hinges on explicit closed-form expressions for $D(G)$0-functions on cycles and paths, leveraging block-diagonal structure, Neumann series expansions, and strict monotonicity of the $D(G)$1-function in $D(G)$2. The increment analysis establishes strict convexity for the path $D(G)$3-functions and bounds on differences between unbalanced partitions, guaranteeing that balanced path lengths minimize the sum.

Balancing Principles and the Exclusion of Cycles

The analysis proves two balancing principles: (1) among partitions of $D(G)$4 into $D(G)$5 paths, the balanced partition (component sizes differ by at most one) yields the minimum $D(G)$6-function; (2) increasing the total order monotonicity increases balanced part sizes. By combining these with increment estimates, configurations involving cycles are shown to be strictly suboptimal—since cycles yield strictly higher $D(G)$7-function values at any admissible $D(G)$8—and thus excluded in all cases except $D(G)$9, where the complete graph is optimal.

Unified Proofs via Walk Enumeration

For the range $s \geq \max\{\frac{n-6}{2}, 1\}$0, complement configurations degenerate to disjoint edges and isolated vertices, permitting walk enumeration. Expanding the $s \geq \max\{\frac{n-6}{2}, 1\}$1-function via the Neumann series, the total number of walks in a graph $s \geq \max\{\frac{n-6}{2}, 1\}$2 is minimized precisely when $s \geq \max\{\frac{n-6}{2}, 1\}$3 is a disjoint union of $s \geq \max\{\frac{n-6}{2}, 1\}$4 edges and $s \geq \max\{\frac{n-6}{2}, 1\}$5 isolated vertices. Any deviation, e.g., the presence of paths longer than two, introduces additional walks and strictly increases $s \geq \max\{\frac{n-6}{2}, 1\}$6.

Numerical and Spectral Arguments

The argument is substantiated by explicit inequalities for increments and differences between $s \geq \max\{\frac{n-6}{2}, 1\}$7-functions, with strict positivity and convexity ensuring that local modifications to path lengths or introduction of cycles can never yield a lower spectral radius. Edge-counting and degree-based arguments are combined with global matrix analysis to confirm uniqueness of the minimizer.

Implications and Future Directions

This complete resolution of the distance spectral extremal problem for graphs with prescribed size $s \geq \max\{\frac{n-6}{2}, 1\}$8 underlines the utility of matrix-analytic and walk-based approaches for extremal spectral graph theory. The techniques developed—particularly the $s \geq \max\{\frac{n-6}{2}, 1\}$9-function comparison and global increment analysis—open pathways for similar treatments in other spectral problems, including adjacency spectral radius, signless Laplacian, and forbidden subgraph regimes. Extensions to color-critical graphs and connections to spectral supersaturation are discussed, suggesting applications to stability-type results and combinatorial optimization in networks.

Conclusion

The paper provides a complete, rigorous resolution of Lin and Zhou's conjecture on distance spectral extremal problems, identifying that the unique minimizer of the distance spectral radius among connected graphs with $\overline{P_{n,s+1}}$0 edges—across the entire admissible range— is the complement of a balanced disjoint union of paths, $\overline{P_{n,s+1}}$1, except in the case $\overline{P_{n,s+1}}$2 where the complete graph $\overline{P_{n,s+1}}$3 prevails. The results are established through global matrix-analytic comparison, increment convexity, explicit spectral calculations, and unified walk enumeration arguments. This framework not only resolves a longstanding open question but also suggests a methodology for tackling broader extremal spectral problems in graph theory.