On a conjecture of Nikiforov concerning the minimal $p$-energy of connected graphs

TL;DR

This paper proves Nikiforov's conjecture that among connected graphs of a given size, the star graph minimizes the p-energy for 1<p<2, using spectral analysis and Coulson-Jacobs-type formulas.

Contribution

The paper fully resolves Nikiforov's conjecture for all connected graphs, extending previous results limited to bipartite graphs, and characterizes the equality condition.

Findings

Star graph minimizes p-energy for 1<p<2 among connected graphs

Equality holds only for the star graph

Complete resolution of Nikiforov's conjecture

Abstract

For a given simple graph \( G \), the \( p \)-energy of \( G \), denoted by \( \mathcal{E}_p(G) \), is defined as the sum of the \( p \)-th power of the absolute values of the eigenvalues of its adjacency matrix. Let \( S_n \) denote the star graph with one internal node and \( n-1 \) leaves. Nikiforov conjectured that for \( 1 < p < 2 \), the connected graph of order \( n \) with the smallest \( p \)-energy is \( S_n \). Recently, this conjecture was proved for bipartite graphs. In this paper, by employing a Coulson-Jacobs-type formula and certain spectral radius results for connected graphs, we completely resolve this conjecture. Furthermore, we establish that the equality condition in the inequality \( \mathcal{E}_p(G) \geq \mathcal{E}_p(S_n) \) holds if and only if \( G \) is \( S_n \).