Characterizing the positive inertia index of connected signed graphs in terms of girth
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915185107664896 |
|---|---|
| author | Khan, Suliman Hayat, Sakander Alenazi, Mohammed J. F. |
| author_facet | Khan, Suliman Hayat, Sakander Alenazi, Mohammed J. F. |
| contents | Let $G^σ=(G,σ)$ be a connected signed graph and $A(G^σ)$ be its adjacency matrix. The positive inertia index of $G^σ$, denoted by $p^{+}(G^σ)$, is defined as the number of positive eigenvalues of $A(G^σ)$. Assume that $G^σ$ contains at least one cycle, and let $g_{r}$ be its girth. In this paper, we prove $p^{+}(G^σ) \geq \lceil \frac {g_{r}}{2} \rceil-1$ for a signed graph $G^σ$. The extremal signed graphs corresponding to $p^{+}(G^σ) = \lceil \frac {g_{r}}{2} \rceil-1$ and $p^{+}(G^σ) =\lceil \frac {g_{r}}{2} \rceil$ are characterized, respectively. The results presented in this article extend the recent work on ordinary graphs by Duan and Yang (Linear Algebra Appl., 2024) to the context of signed graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05135 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing the positive inertia index of connected signed graphs in terms of girth Khan, Suliman Hayat, Sakander Alenazi, Mohammed J. F. Combinatorics 05C22, 05C50 Let $G^σ=(G,σ)$ be a connected signed graph and $A(G^σ)$ be its adjacency matrix. The positive inertia index of $G^σ$, denoted by $p^{+}(G^σ)$, is defined as the number of positive eigenvalues of $A(G^σ)$. Assume that $G^σ$ contains at least one cycle, and let $g_{r}$ be its girth. In this paper, we prove $p^{+}(G^σ) \geq \lceil \frac {g_{r}}{2} \rceil-1$ for a signed graph $G^σ$. The extremal signed graphs corresponding to $p^{+}(G^σ) = \lceil \frac {g_{r}}{2} \rceil-1$ and $p^{+}(G^σ) =\lceil \frac {g_{r}}{2} \rceil$ are characterized, respectively. The results presented in this article extend the recent work on ordinary graphs by Duan and Yang (Linear Algebra Appl., 2024) to the context of signed graphs. |
| title | Characterizing the positive inertia index of connected signed graphs in terms of girth |
| topic | Combinatorics 05C22, 05C50 |
| url | https://arxiv.org/abs/2503.05135 |