Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Walks, infinite series and spectral radius of graphs

Wenqian Zhang

TL;DR

This work addresses how the spectral radius $\rho(G)$ of a finite graph relates to the walks in its subgraphs, particularly when $G$ contains a spanning complete multipartite subgraph. It develops an infinite-series framework based on walk counts $W^{\ell}(G)$ to compare spectral radii after embedding edge-subgraphs into parts, and proves a root-finding criterion for $\rho$ in these configurations. A key transfer principle shows that subgraph-walk order ($H_1\equiv H_2$, $H_1\succ H_2$, or $H_1\prec H_2$) induces the corresponding spectral-order for large $\rho$, enabling a structural SPEX analysis. The results yield a precise characterization of extremal graphs in Turán-type families via a per-part subgraph $H^G$ lying in ${\rm EX}^{\infty}(\mathcal{H})$ and a computable equation for the spectral radius, advancing spectral extremal problems.

Abstract

For a graph G, the spectral radius \r{ho}(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, we seek the relationship between \r{ho}(G) and the walks of the subgraphs of G. Especially, if G contains a complete multi-partite graph as a spanning subgraph, we give a formula for \r{ho}(G) by using an infinite series on walks of the subgraphs of G. These results are useful for the current popular spectral extremal problem.

Walks, infinite series and spectral radius of graphs

TL;DR

This work addresses how the spectral radius of a finite graph relates to the walks in its subgraphs, particularly when contains a spanning complete multipartite subgraph. It develops an infinite-series framework based on walk counts to compare spectral radii after embedding edge-subgraphs into parts, and proves a root-finding criterion for in these configurations. A key transfer principle shows that subgraph-walk order (, , or ) induces the corresponding spectral-order for large , enabling a structural SPEX analysis. The results yield a precise characterization of extremal graphs in Turán-type families via a per-part subgraph lying in and a computable equation for the spectral radius, advancing spectral extremal problems.

Abstract

For a graph G, the spectral radius \r{ho}(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, we seek the relationship between \r{ho}(G) and the walks of the subgraphs of G. Especially, if G contains a complete multi-partite graph as a spanning subgraph, we give a formula for \r{ho}(G) by using an infinite series on walks of the subgraphs of G. These results are useful for the current popular spectral extremal problem.
Paper Structure (3 sections, 10 theorems, 41 equations)

This paper contains 3 sections, 10 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorem and Theorem
  3. Proof of Theorem

Key Result

Theorem 1.1

(Wang, Kang and Xue WKX) For $r\geq2$ and sufficiently large $n$, let $\mathcal{H}$ be a finite family of graphs such that all graphs in ${\rm EX}(n,\mathcal{H})$ are obtained from $T_{n,r}$ by embedding $k$ edges into its $r$ parts, where $k$ is a bounded integer. Then ${\rm SPEX}(n,\mathcal{H})\su

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3