On a conjecture of Nikiforov concerning the minimal p-energy of connected graphs
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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 \( S_n \).
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Cited by 3 Pith papers
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Path-Minimality of $p$-Energy for Connected Graphs
For p ≥ 2 the p-energy of any connected graph on n vertices is minimized by the path P_n, with equality only for the path when p > 2.
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Path-Minimality of $p$-Energy for Connected Graphs
The path graph minimizes p-energy among connected graphs for p ≥ 2.
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Path-Minimality of $p$-Energy for Connected Graphs
For every p ≥ 2 and every connected simple graph G on n vertices, the p-energy E_p(G) is at least E_p(P_n), with equality for p > 2 if and only if G is the path.
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