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The distance spectral radius of $k$-uniform hypertrees with given number of vertices of maximum degree

Xiaoqi Liu, Haiying Shan

Abstract

This paper investigates the influence of two graft transformations on the distance spectral radius of connected uniform hypergraphs. Specifically, we study $k$-uniform hypertrees with given size, maximum degree and number of vertices of maximum degree, and give the structure of such hypergraph with maximum distance spectral radius.

The distance spectral radius of $k$-uniform hypertrees with given number of vertices of maximum degree

Abstract

This paper investigates the influence of two graft transformations on the distance spectral radius of connected uniform hypergraphs. Specifically, we study -uniform hypertrees with given size, maximum degree and number of vertices of maximum degree, and give the structure of such hypergraph with maximum distance spectral radius.
Paper Structure (4 sections, 15 theorems, 62 equations, 1 figure)

This paper contains 4 sections, 15 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Distance spectral radius of caterpillars in $\mathbb{T}_k(m,\Delta,n)$
  4. Distance spectral radius of trees in $\mathbb{T}_k(m,\Delta,n)$

Key Result

Lemma 2.1

lin2 Let $H$ be a connected $k$-uniform hypergraph with $|E(H)| \geq 1$ and $u \in V(H).$ For integers $s\geq t\geq 1$, $\rho( H_u(s, t)) <\rho (H_u(s+1, t-1)).$

Figures (1)

  • Figure 1: $H_{u,v}(P_s,P_t,G_r)$

Theorems & Definitions (27)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.1
  • Corollary 2.2
  • Lemma 2.6
  • ...and 17 more