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Extremal spectral radius of degree-based weighted adjacency matrices of graphs with given order and size

Chenghao Shen, Haiying Shan

Abstract

The $f$ adjacency matrix is a type of edge-weighted adjacency matrix, whose weight of an edge $ij$ is $f(d_i,d_j)$, where $f$ is a real symmetric function and $d_i,d_j$ are the degrees of vertex $i$ and vertex $j$. The $f$-spectral radius of a graph is the spectral radius of its $f$-adjacency matrix. In this paper, the effect of subdividing an edge on $f$-spectral radius is discussed. Some necessary conditions of the extremal graph with given order and size are derived. As an example, we obtain the bicyclic graph(s) with the smallest $f$-spectral radius for fixed order $n\geq8$ by applying generalized Lu-Man method.

Extremal spectral radius of degree-based weighted adjacency matrices of graphs with given order and size

Abstract

The adjacency matrix is a type of edge-weighted adjacency matrix, whose weight of an edge is , where is a real symmetric function and are the degrees of vertex and vertex . The -spectral radius of a graph is the spectral radius of its -adjacency matrix. In this paper, the effect of subdividing an edge on -spectral radius is discussed. Some necessary conditions of the extremal graph with given order and size are derived. As an example, we obtain the bicyclic graph(s) with the smallest -spectral radius for fixed order by applying generalized Lu-Man method.
Paper Structure (6 sections, 45 equations)

This paper contains 6 sections, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subdividing an edge
  4. Applying generalized Lu-Man method on an internal path
  5. The $c$-cyclic graph with the smallest $\rho_f$
  6. The $c$-cyclic graph with the largest $\rho_f$

Theorems & Definitions (15)

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  • ...and 5 more