Path-Minimality of $p$-Energy for Connected Graphs
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arXiv
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2026
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| _version_ | 1866913153286144000 |
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| author | Liu, Yinchen Tang, Quanyu |
| author_facet | Liu, Yinchen Tang, Quanyu |
| contents | Let $G$ be a simple connected graph on $n$ vertices, and let $λ_1(G),λ_2(G),\ldots,λ_n(G)$ be the eigenvalues of its adjacency matrix $A(G)$. For $p>0$, define the $p$-energy of $G$ by $\mathcal E_p(G)=\sum_{i=1}^n |λ_i(G)|^p$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_22730 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Path-Minimality of $p$-Energy for Connected Graphs Liu, Yinchen Tang, Quanyu Combinatorics Primary 05C50, Secondary 05C35, 15A18 Let $G$ be a simple connected graph on $n$ vertices, and let $λ_1(G),λ_2(G),\ldots,λ_n(G)$ be the eigenvalues of its adjacency matrix $A(G)$. For $p>0$, define the $p$-energy of $G$ by $\mathcal E_p(G)=\sum_{i=1}^n |λ_i(G)|^p$. 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. |
| title | Path-Minimality of $p$-Energy for Connected Graphs |
| topic | Combinatorics Primary 05C50, Secondary 05C35, 15A18 |
| url | https://arxiv.org/abs/2605.22730 |