Path-Minimality of $p$-Energy for Connected Graphs

TL;DR

This paper proves that among all connected graphs on n vertices, the path minimizes the p-energy for p ≥ 2, providing a complete solution to related extremal spectral graph theory questions.

Contribution

It establishes the path as the minimal graph for p-energy when p ≥ 2, completing the solution to two open questions of Nikiforov and introducing new comparison techniques.

Findings

Paths minimize p-energy for p ≥ 2 in connected graphs.

Equality holds only for paths when p > 2.

Results extend to Laplacian and signless Laplacian power sums.

Abstract

Let $G$ be a simple connected graph on $n$ vertices, and let $\lambda_1(G),\lambda_2(G),\ldots,\lambda_n(G)$ be the eigenvalues of its adjacency matrix $A(G)$. For $p>0$, define the $p$-energy of $G$ by $\mathcal E_p(G)=\sum_{i=1}^n |\lambda_i(G)|^p$. We prove that, for every real number $p\ge 2$ and every simple connected graph $G$ on $n$ vertices, $$ \mathcal E_p(G)\ge \mathcal E_p(P_n), $$ where $P_n$ denotes the path on $n$ vertices. Moreover, for each fixed $p>2$, equality holds if and only if $G\cong P_n$. Together with the previously known star-minimality results, this completes the solution of two questions of Nikiforov. The proof combines two different comparison principles. For $2<p<4$, we use a bipartite reduction, a Mellin representation of fractional powers, and a determinant comparison involving matching generating polynomials and tree shifts. For $p\ge4$, we prove a second-order stop-loss comparison for the squared singular values of bipartite graphs. This comparison is established by rank-one spectral-shift estimates, deletion-minimal counterexamples, and a finite certified analysis of the terminal sparse-sun configurations. As applications, we obtain sharp path-minimality results for positive $p$-energies in several cases, and for Laplacian and signless Laplacian power sums and related indices.