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Spectral conditions for graphs in which every edge belongs to a factor

Jin Cai, Bo Zhou

Abstract

A factor of a graph is a spanning subgraph. Spectral sufficient conditions are provided via spectral radius and signless Laplacian spectral radius for graphs with (i) a matching of given size (particularly, $1$-factor) containing any given edge, and (ii) a star factor with a component isomorphic to stars of order two or three containing any given edge, respectively.

Spectral conditions for graphs in which every edge belongs to a factor

Abstract

A factor of a graph is a spanning subgraph. Spectral sufficient conditions are provided via spectral radius and signless Laplacian spectral radius for graphs with (i) a matching of given size (particularly, -factor) containing any given edge, and (ii) a star factor with a component isomorphic to stars of order two or three containing any given edge, respectively.
Paper Structure (8 sections, 18 theorems, 47 equations)

This paper contains 8 sections, 18 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs with a matching and $1$-factor containing any given edge
  4. Spectral radius
  5. Signless Laplacian spectral radius
  6. Star factor with a component $K_{1,1}$ or $K_{1,2}$ containing any given edge
  7. Spectral radius
  8. Signless Laplacian spectral radius

Key Result

Theorem 1.1

Let $G$ be a graph of order $n\ge 5k+6$, where $n \equiv k \pmod 2$. If $\rho(G)\ge \rho(K_2\vee ((k+1)K_1\cup K_{n-k-3}))$, then $G$ has a matching of size $\frac{n-k}{2}$ containing any given edge unless $G\cong K_2\vee ((k+1)K_1\cup K_{n-k-3})$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 19 more