PreprintCC BY 4.0arXiv
A Vertex-Localized Positive Square-Energy Strengthening of Turán's Theorem
Abstract
Let G be a graph of order n with the adjacency eigenvalues $λ_1(G) \geq \dots \geq λ_n(G) $. Let c(v) denote the maximum order of a clique containing vertex v. We prove the vertex-localized positive square-energy inequality √(s+(G)) ≤ ∑v∈ V(1−1/c(v)), where s+(G)=∑λi(G)>0λi(G)2. We also characterize equality. Apart from edgeless graphs, equality holds precisely for graphs obtained from a complete regular multipartite graph by adding an arbitrary number of isolated vertices. This settles a conjecture of Kannan, Kumar and Pragada.
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BibTeX
@misc{jayarajan2026vertexlocalized,
title = {{A Vertex-Localized Positive Square-Energy Strengthening of Turán's Theorem}},
author = {Abhay Jayarajan and M. Rajesh Kannan and Shivaramakrishna Pragada and Rahul Roy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24861},
eprint = {2608.24861},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24861},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Vertex-Localized Positive Square-Energy Strengthening of Turán's Theorem
AU - Abhay Jayarajan
AU - M. Rajesh Kannan
AU - Shivaramakrishna Pragada
AU - Rahul Roy
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.24861
UR - https://arxiv.org/abs/2608.24861
AB - Let G be a graph of order n with the adjacency eigenvalues $λ_1(G) \geq \dots \geq λ_n(G) $. Let c(v) denote the maximum order of a clique containing vertex v. We prove the vertex-localized positive square-energy inequality √(s_+(G)) ≤ ∑_v∈ V(1−1/c(v)), where s_+(G)=∑_λ_i(G)>0λ_i(G)^2. We also characterize equality. Apart from edgeless graphs, equality holds precisely for graphs obtained from a complete regular multipartite graph by adding an arbitrary number of isolated vertices. This settles a conjecture of Kannan, Kumar and Pragada.
ER -