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Bound for the energy of graphs in terms of degrees and leaves

Octavio Arizmendi, Samuel Gurrola-Viramontes

TL;DR

This paper addresses the problem of bounding graph energy by exploiting local structural features—vertex degrees and leaves. It introduces a novel upper bound $\mathcal{E}(G) \le 2 e_{11} + \sum_{v \in V'} \sqrt{3 l(v) + d(v)}$ derived from a weighted-star decomposition and Ky-Fan's inequality, with a global version and comparisons to known bounds. The authors apply the bound to Barabasi–Albert trees and sparse Erdős–Rényi graphs, obtaining sharp asymptotic energy limits: for BA trees, $\limsup_{n\to\infty} \mathcal{E}(T_n)/n \le 0.95999$, and for ER graphs, $\limsup_{n\to\infty} \mathcal{E}(G_n)/n \le f(\lambda)$ with an explicit $f(\lambda)$, yielding hypoenergeticity when $\lambda \le 4/3$. These results provide tighter, structurally grounded energy estimates in sparse random graph regimes and illustrate the practical impact of leaf-degree features on spectral bounds.

Abstract

We provide a new upper bound for the energy of graphs in terms of degrees and number of leaves. We apply this formula to study the energy of Erdös-Rényi graphs and Barabasi-Albert trees.

Bound for the energy of graphs in terms of degrees and leaves

TL;DR

This paper addresses the problem of bounding graph energy by exploiting local structural features—vertex degrees and leaves. It introduces a novel upper bound derived from a weighted-star decomposition and Ky-Fan's inequality, with a global version and comparisons to known bounds. The authors apply the bound to Barabasi–Albert trees and sparse Erdős–Rényi graphs, obtaining sharp asymptotic energy limits: for BA trees, , and for ER graphs, with an explicit , yielding hypoenergeticity when . These results provide tighter, structurally grounded energy estimates in sparse random graph regimes and illustrate the practical impact of leaf-degree features on spectral bounds.

Abstract

We provide a new upper bound for the energy of graphs in terms of degrees and number of leaves. We apply this formula to study the energy of Erdös-Rényi graphs and Barabasi-Albert trees.

Paper Structure

This paper contains 12 sections, 18 theorems, 115 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions on graphs
  4. Some families of graphs
  5. Sachs' theorem for weighted graphs
  6. Inequalities for the Energy of a Graph
  7. Main Result
  8. An inequality for the energy in terms of degrees and leaves
  9. Comparison with previous bounds
  10. Energy for Barabási-Albert Trees and ER graphs
  11. Barabási-Albert Trees
  12. Energy for Erdös-Rényi Graphs

Key Result

theorem 1

Let $G$ be a graph in $n$ vertices and $\phi_G(x) = \sum_{k = 0}^n b_k x^{n-k}$ be its characteristic polynomial. The coefficients $(b_k)_{k \geq 1}$ satisfies the equality where $\mathcal{S}_k(G)$ denotes the set of sub-graphs of $G$ with exactly $k$ vertices that are Sachs' graphs. Furthermore, $b_0 = 1$.

Figures (7)

  • Figure 1: Some examples of path graphs.
  • Figure 2: Some examples of a star graph.
  • Figure 3: Some examples of a cycle graphs.
  • Figure 4: Some examples of connected graphs where the equality between local and global bound holds.
  • Figure 5: Energy / Size for 200 random trees of size $n = 2000$ following the Barabási-Albert model compared with bounds from Proposition \ref{['eq:AD']}, Theorem \ref{['TP']}, McClleland's bound, Corollary \ref{['global_bound']} and the asymptotic bound from Theorem \ref{['Asym_BA']}.
  • ...and 2 more figures

Theorems & Definitions (34)

  • definition 1: Sachs' graph
  • theorem 1: Sachs' theorem
  • definition 2: Sachs' sub-graph for a weighted graph
  • theorem 2: Sach's theorem for weighted graphs
  • proposition 1: Arizmendi and Juarez Arizmendi18
  • proposition 2: Arizmendi and Dominguez Arizmendi22
  • theorem 3: Ky-Fan's theorem for weighted sub-graphs
  • lemma 1
  • proof
  • theorem 4
  • ...and 24 more