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| Format: | Preprint |
| Published: |
2020
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| Online Access: | https://arxiv.org/abs/2011.10270 |
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| _version_ | 1866916383756910592 |
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| author | Batal, Ahmet |
| author_facet | Batal, Ahmet |
| contents | For a simple graph $G$ with vertex set $V(G)=\{v_1,...,v_n\}$, we define the closed neighborhood set of a vertex $u$ as $N[u]=\{v \in V(G) \; | \; v \; \text{is adjacent to} \; u \; \text{or} \; v=u \}$ and the closed neighborhood matrix $N(G)$ as the matrix obtained by setting to $1$ all the diagonal entries of the adjacency matrix of $G$. We say a set $S$ is odd dominating if $N[u]\cap S$ is odd for all $u\in V(G)$. We prove that the parity of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where the rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_10270 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Parity of an odd dominating set Batal, Ahmet Combinatorics 05C69 For a simple graph $G$ with vertex set $V(G)=\{v_1,...,v_n\}$, we define the closed neighborhood set of a vertex $u$ as $N[u]=\{v \in V(G) \; | \; v \; \text{is adjacent to} \; u \; \text{or} \; v=u \}$ and the closed neighborhood matrix $N(G)$ as the matrix obtained by setting to $1$ all the diagonal entries of the adjacency matrix of $G$. We say a set $S$ is odd dominating if $N[u]\cap S$ is odd for all $u\in V(G)$. We prove that the parity of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where the rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs. |
| title | Parity of an odd dominating set |
| topic | Combinatorics 05C69 |
| url | https://arxiv.org/abs/2011.10270 |