On a conjecture of Nikiforov concerning the minimal $p$-energy of connected graphs
Quanyu Tang, Yinchen Liu, Wei Wang
TL;DR
This work settles Nikiforov's conjecture for the minimal $p$-energy of connected graphs in the regime $0<p<2$, showing that $\mathcal{E}_p(G) \ge \mathcal{E}_p(S_n)$ with equality only for the star $S_n$. The authors develop a Coulson-Jacobs-type integral formula for $p$-energy differences and sharp spectral-radius bounds to compare energies, then prove the conjecture by establishing a pointwise lower bound on the modulus of the characteristic polynomial evaluated at purely imaginary arguments. The key step is showing $|f(ix)| \ge |x^{n-2}(x^2+n-1)|$ for all $x\ge0$, together with the $\mathcal{E}_4$ upper bound, which together force equality only at $G=S_n$. These results complete the resolution for all connected graphs in this range and outline potential avenues and obstacles for extending the approach to $p>2$ and to identifying the exact equality cases there.
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 \( S_n \).