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PreprintCC0arXiv

Positive and Negative Square Energies of $2$-Connected Graphs

  • S. Akbari
  • Fu-Tao Hu
  • Ya-Yang Liu
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Abstract

Let G be a graph of order n, and let s+(G) and s-(G) denote the sums of the squares of the positive and negative adjacency eigenvalues of G, respectively. Recently, Liu, Tang, and Zhang proved the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph G of order n satisfies $ \min\{s^+(G), s^-(G)\} \ge n-1. $ For positive square energy, we strengthen this result by showing that every $2$-connected graph G of order n which is not a cycle satisfies s+(G)≥ n. The formally analogous assertion for s- is false: the complete graph Kn satisfies s-(Kn)=n−1. We prove a natural counterpart in the triangle-free class: every triangle-free $2$-connected noncycle G satisfies $ \min\{s^+(G),s^-(G)\}>n. $ More generally, it is enough that some maximum-degree vertex of G belongs to no triangle. Together with the exact square energies of cycles, this characterizes the triangle-free $2$-connected graphs for which s-(G)≥ n; the only exceptions are the cycles C4k+3 with k≥1.

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Cite

BibTeXRIS
BibTeX
@misc{akbari2026positive,
  title = {{Positive and Negative Square Energies of $2$-Connected Graphs}},
  author = {S. Akbari and Fu-Tao Hu and Ya-Yang Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04069},
  eprint = {2609.04069},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04069},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Positive and Negative Square Energies of $2$-Connected Graphs
AU  - S. Akbari
AU  - Fu-Tao Hu
AU  - Ya-Yang Liu
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04069
UR  - https://arxiv.org/abs/2609.04069
AB  - Let G be a graph of order n, and let s^+(G) and s^-(G) denote the sums of the squares of the positive and negative adjacency eigenvalues of G, respectively. Recently, Liu, Tang, and Zhang proved the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph G of order n satisfies $ \min\{s^+(G), s^-(G)\} \ge n-1. $ For positive square energy, we strengthen this result by showing that every $2$-connected graph G of order n which is not a cycle satisfies s^+(G)≥ n. The formally analogous assertion for s^- is false: the complete graph K_n satisfies s^-(K_n)=n−1. We prove a natural counterpart in the triangle-free class: every triangle-free $2$-connected noncycle G satisfies $ \min\{s^+(G),s^-(G)\}>n. $ More generally, it is enough that some maximum-degree vertex of G belongs to no triangle. Together with the exact square energies of cycles, this characterizes the triangle-free $2$-connected graphs for which s^-(G)≥ n; the only exceptions are the cycles C_4k+3 with k≥1.
ER  -