Path-Minimality for Positive $p$-Energies, Laplacian-Type Spectra, and Line Graphs

Abstract

We derive several applications of the path-minimality theorem for adjacency $p$-energy proved in the companion paper. First, we prove the sharp inequality $${\mathcal{E}}_p^{+}(G)\ge {\mathcal{E}}_p^{+}(P_n),$$ where $P_n$ is the path on $n$ vertices, in three settings: connected bipartite graphs for every real $p\ge 2$, all connected graphs for every odd integer $p\ge 3$, and all connected graphs for $p=4$. Second, using subdivision graphs, we prove path-minimality for Laplacian and signless Laplacian-type spectral sums, including power sums, Estrada-type quantities, resolvent energies, and thresholded tails. Third, we prove an edge-count second-order stop-loss comparison for the signless Laplacian above the threshold $2$. This yields the sharp line-graph inequality $${\mathcal{E}}_p^{+}(\mathcal{L}(G))\ge {\mathcal{E}}_p^{+}(P_m)$$ for every connected graph $G$ with $m$ edges and every real $p\ge 2$.

Standalone companion paper split from arXiv:2605.22730v1. This paper contains the applications and further consequences formerly included in the comprehensive v1 version; the revised focused main paper appears as arXiv:2605.22730v2