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

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 ${\mathcal{E}}_p(G)\ge {\mathcal{E}}_p(S_n)$ holds if and only if $G$ is $S_n$.