A Linear Lower Bound for the Square Energy of Graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913520699834368 |
|---|---|
| author | Akbari, Saieed Kumar, Hitesh Mohar, Bojan Pragada, Shivaramakrishna |
| author_facet | Akbari, Saieed Kumar, Hitesh Mohar, Bojan Pragada, Shivaramakrishna |
| contents | Let $G$ be a graph of order $n$ with eigenvalues $λ_1 \geq \cdots \geqλ_n$. Let \[s^+(G)=\sum_{λ_i>0} λ_i^2, \qquad s^-(G)=\sum_{λ_i<0} λ_i^2.\] The smaller value, $s(G)=\min\{s^+(G), s^-(G)\}$ is called the \emph{square energy} of $G$. In 2016, Elphick, Farber, Goldberg and Wocjan conjectured that for every connected graph $G$ of order $n$, $s(G)\geq n-1.$ No linear bound for $s(G)$ in terms of $n$ is known. Let $H_1, \ldots, H_k$ be disjoint vertex-induced subgraphs of $G$. In this note, we prove that \[s^+(G)\geq\sum_{i=1}^{k} s^+(H_i) \quad \text{ and } \quad s^-(G)\geq\sum_{i=1}^{k} s^-(H_i),\] which implies that $s(G)\geq \frac{3n}{4}$ for every connected graph $G$ of order $n\ge 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18220 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Linear Lower Bound for the Square Energy of Graphs Akbari, Saieed Kumar, Hitesh Mohar, Bojan Pragada, Shivaramakrishna Combinatorics 05C50 Let $G$ be a graph of order $n$ with eigenvalues $λ_1 \geq \cdots \geqλ_n$. Let \[s^+(G)=\sum_{λ_i>0} λ_i^2, \qquad s^-(G)=\sum_{λ_i<0} λ_i^2.\] The smaller value, $s(G)=\min\{s^+(G), s^-(G)\}$ is called the \emph{square energy} of $G$. In 2016, Elphick, Farber, Goldberg and Wocjan conjectured that for every connected graph $G$ of order $n$, $s(G)\geq n-1.$ No linear bound for $s(G)$ in terms of $n$ is known. Let $H_1, \ldots, H_k$ be disjoint vertex-induced subgraphs of $G$. In this note, we prove that \[s^+(G)\geq\sum_{i=1}^{k} s^+(H_i) \quad \text{ and } \quad s^-(G)\geq\sum_{i=1}^{k} s^-(H_i),\] which implies that $s(G)\geq \frac{3n}{4}$ for every connected graph $G$ of order $n\ge 4$. |
| title | A Linear Lower Bound for the Square Energy of Graphs |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2409.18220 |