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Some lemmas on spectral radius of graphs: including an application

Wenqian Zhang

TL;DR

This work analyzes the spectral radius $ρ(G)$ of finite graphs containing a spanning complete bipartite subgraph, proving three comparison lemmas that relate $ρ(G)$ to how path-like components are arranged in joins with a fixed graph. It develops a walk-count framework, introducing $W^{\ell}(G)$ and a multi-partite spectral equation, to establish monotonicity results under redistribution of path lengths. An application to $C_{\ell}$-free planar graphs yields sharp extremal structures for large $n$, giving explicit forms for the maximizing graphs in two regimes. Overall, the paper advances extremal spectral graph theory in planar graphs by reducing the problem to join-with-path constructions and providing precise extremal configurations.

Abstract

For a graph $G$, the spectral radius $ρ(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we give three lammas on $ρ(G)$ when $G$ contains a spanning complete bipartite graph. Moreover, an application was also included at the end.

Some lemmas on spectral radius of graphs: including an application

TL;DR

This work analyzes the spectral radius of finite graphs containing a spanning complete bipartite subgraph, proving three comparison lemmas that relate to how path-like components are arranged in joins with a fixed graph. It develops a walk-count framework, introducing and a multi-partite spectral equation, to establish monotonicity results under redistribution of path lengths. An application to -free planar graphs yields sharp extremal structures for large , giving explicit forms for the maximizing graphs in two regimes. Overall, the paper advances extremal spectral graph theory in planar graphs by reducing the problem to join-with-path constructions and providing precise extremal configurations.

Abstract

For a graph , the spectral radius of is the largest eigenvalue of its adjacency matrix. In this paper, we give three lammas on when contains a spanning complete bipartite graph. Moreover, an application was also included at the end.
Paper Structure (4 sections, 9 theorems, 47 equations, 1 figure)

This paper contains 4 sections, 9 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of Lemmas \ref{['1-path']}, \ref{['2-path']} and \ref{['3-path']}
  4. Spectral radius of $C_{\ell}$-free planar graphs

Key Result

Lemma 1.1

For integers $n_{1}\geq n_{2}+2\geq1$, let $G_{1}=H\vee(P_{n_{1}}\cup P_{n_{2}}\cup T)$ and $G_{2}=H\vee(P_{n_{1}-1}\cup P_{n_{2}+1}\cup T)$, where $H,T$ are two graphs with $|H|\geq1$ and $|T|\geq0$. Then $\rho(G_{1})>\rho(G_{2})$.

Figures (1)

  • Figure 1: The infinite path $P_{\infty}$

Theorems & Definitions (9)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 4.1