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DQ-integral and DL-integral generalized wheel graphs

Yirui Chai, Ligong Wang, Yuwei Zhou

Abstract

A graph G is said to be M-integral (resp. A-integral, D-integral, DL-integral or DQ-integral) if all eigenvalues of its matrix M (resp. adjacency matrix A(G), distance matrix D(G), distance Laplacian matrix DL(G) or distance signless Laplacian matrix DQ(G)) are integers. Lu et al. [Discrete Math, 346 (2023)] defined the generalized wheel graph GW(a, m, n) as the join of two regular graphs aKm and Cn, and obtained all D-integral generalized wheel graphs. Based on the above research, in this paper, we determine all DL-integral and DQ-integral generalized wheel graphs respectively. As byproducts, we give a sufficient and necessary condition for the join of two regular graphs G1 and G2 to be DL-integral, from which we can get infinitely many new classes of DL-integral graphs according to the large number of research results about the A-integral graphs.

DQ-integral and DL-integral generalized wheel graphs

Abstract

A graph G is said to be M-integral (resp. A-integral, D-integral, DL-integral or DQ-integral) if all eigenvalues of its matrix M (resp. adjacency matrix A(G), distance matrix D(G), distance Laplacian matrix DL(G) or distance signless Laplacian matrix DQ(G)) are integers. Lu et al. [Discrete Math, 346 (2023)] defined the generalized wheel graph GW(a, m, n) as the join of two regular graphs aKm and Cn, and obtained all D-integral generalized wheel graphs. Based on the above research, in this paper, we determine all DL-integral and DQ-integral generalized wheel graphs respectively. As byproducts, we give a sufficient and necessary condition for the join of two regular graphs G1 and G2 to be DL-integral, from which we can get infinitely many new classes of DL-integral graphs according to the large number of research results about the A-integral graphs.
Paper Structure (3 sections, 15 theorems, 32 equations)

This paper contains 3 sections, 15 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. $D^{Q}$-integral generalized wheel graphs $aK_{m} \nabla C_{n}$
  3. Results on $D^{L}$--integrality for the join of regular graphs

Key Result

Lemma 2.1

(DaDeDe) For $i = 1, 2$, let $G_{i}$ be an $r_{i}$-regular graph with $n_{i}$ vertices. If the eigenvalues of the adjacency matrix of $G_{i}$ are given by $r_{i}=\lambda_{1}^{(i)}\geq \lambda_{2}^{(i)} \geq \cdots \geq \lambda_{n_{i}}^{(i)}$, then the distance signless Laplacian spectrum of $G_{1} \

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 5 more