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

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 $p\ge2$, all connected graphs for every odd integer $p\ge3$, 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 $p\ge2$.