Inertia indices of signed graphs with given cyclomatic number and given number of pendant vertices
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909665378435072 |
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| author | Pu, Jie Duan, Fang |
| author_facet | Pu, Jie Duan, Fang |
| contents | Let $Γ=(G, σ)$ be a signed graph of order $n$ with underlying graph $G$ and a sign function $σ: E(G)\rightarrow \{+, -\}$. Denoted by $i_+(Γ)$, $θ(Γ)$ and $p(Γ)$ the positive inertia index, the cyclomatic number and the number of pendant vertices of $Γ$, respectively. In this article, we prove that $i_+(Γ)$, $θ(Γ)$ and $p(Γ)$ are related by the inequality $i_+(Γ)\geq \frac{n-p(Γ)}{2}-θ(Γ)$. Furthermore, we completely characterize the signed graph $Γ$ for which $i_+(Γ)=\frac{n-p(Γ)}{2}-θ(Γ)$. As a by-product, the inequalities $i_-(Γ)\geq \frac{n-p(Γ)}{2}-θ(Γ)$ and $η(Γ)\leq p(Γ)+2θ(Γ)$ are also obtained, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23112 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inertia indices of signed graphs with given cyclomatic number and given number of pendant vertices Pu, Jie Duan, Fang Spectral Theory Statistics Theory Let $Γ=(G, σ)$ be a signed graph of order $n$ with underlying graph $G$ and a sign function $σ: E(G)\rightarrow \{+, -\}$. Denoted by $i_+(Γ)$, $θ(Γ)$ and $p(Γ)$ the positive inertia index, the cyclomatic number and the number of pendant vertices of $Γ$, respectively. In this article, we prove that $i_+(Γ)$, $θ(Γ)$ and $p(Γ)$ are related by the inequality $i_+(Γ)\geq \frac{n-p(Γ)}{2}-θ(Γ)$. Furthermore, we completely characterize the signed graph $Γ$ for which $i_+(Γ)=\frac{n-p(Γ)}{2}-θ(Γ)$. As a by-product, the inequalities $i_-(Γ)\geq \frac{n-p(Γ)}{2}-θ(Γ)$ and $η(Γ)\leq p(Γ)+2θ(Γ)$ are also obtained, respectively. |
| title | Inertia indices of signed graphs with given cyclomatic number and given number of pendant vertices |
| topic | Spectral Theory Statistics Theory |
| url | https://arxiv.org/abs/2506.23112 |