Counterexamples to a conjecture on graph inertia
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917471479398400 |
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| author | Chen, Hongzhang Li, Jianxi |
| author_facet | Chen, Hongzhang Li, Jianxi |
| contents | The inertia of a graph $G$ is $\operatorname{In}(G)=(n^+(G),n^0(G),n^-(G))$, where $n^+(G),\, n^0(G),\, n^-(G)$ are the numbers of positive, zero and negative eigenvalues of the adjacency matrix of $G$, respectively, counted with multiplicities. Akbari, Elphick, Kumar, Pragada and Tang [Discrete Math. 349 (2026) 114953] conjectured that every graph $G$ satisfies \[
2n^+(G)\le n^-(G)(n^-(G)+1). \] In this note, we construct a family of reduced graphs $\{W_{k}:\,k\ge5\}$ with \[
\operatorname{In}(W_k)
=
\left(\binom{k}{2}+1,\ 0,\ k-1\right), \] each of which violates the conjectured inequality. We also observe that deleting the vertex $a_1$ from $W_5$ gives a reduced graph with inertia $(10,0,4)$, answering a question raised in the same paper. The family also refutes a weaker inequality proposed there. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07196 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counterexamples to a conjecture on graph inertia Chen, Hongzhang Li, Jianxi Combinatorics 05C50 The inertia of a graph $G$ is $\operatorname{In}(G)=(n^+(G),n^0(G),n^-(G))$, where $n^+(G),\, n^0(G),\, n^-(G)$ are the numbers of positive, zero and negative eigenvalues of the adjacency matrix of $G$, respectively, counted with multiplicities. Akbari, Elphick, Kumar, Pragada and Tang [Discrete Math. 349 (2026) 114953] conjectured that every graph $G$ satisfies \[ 2n^+(G)\le n^-(G)(n^-(G)+1). \] In this note, we construct a family of reduced graphs $\{W_{k}:\,k\ge5\}$ with \[ \operatorname{In}(W_k) = \left(\binom{k}{2}+1,\ 0,\ k-1\right), \] each of which violates the conjectured inequality. We also observe that deleting the vertex $a_1$ from $W_5$ gives a reduced graph with inertia $(10,0,4)$, answering a question raised in the same paper. The family also refutes a weaker inequality proposed there. |
| title | Counterexamples to a conjecture on graph inertia |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2605.07196 |