The unitary Cayley graph of upper triangular matrix rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913275605680128 |
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| author | Hołubowski, Waldemar Kozerenko, Sergiy Oliynyk, Bogdana Solomko, Viktoriia |
| author_facet | Hołubowski, Waldemar Kozerenko, Sergiy Oliynyk, Bogdana Solomko, Viktoriia |
| contents | The unitary Cayley graph $C_R$ of a finite unital ring $R$ is the simple graph with vertex set $R$ in which two elements $x$ and $y$ are connected by an edge if and only if $x-y$ is a unit of $R$. We characterize the unitary Cayley graph $C_{T_n (\mathbb{F})}$ of the ring of all upper triangular matrices $T_n(\mathbb{F})$ over a finite field $\mathbb{F}$. We show that $C_{T_n (\mathbb{F})}$ is isomorphic to the semistrong product of the complete graph $K_m$ and the antipodal graph of the Hamming graph $A(H(n,p^k))$, where $m=p^{\frac{kn(n-1)}{2}}$ and $|\mathbb{F}|=p^k$. In particular, if $|\mathbb{F}|=2$, then the graph $C_{T_n (\mathbb{F})}$ has $2^{n-1}$ connected components, each component is isomorphic to the complete bipartite graph $K_{m,m}$, where $m=2^{\frac{n(n-1)}{2}}$. We also compute the diameter, triameter, and clique number of the graph $C_{T_n (\mathbb{F})}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_01303 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The unitary Cayley graph of upper triangular matrix rings Hołubowski, Waldemar Kozerenko, Sergiy Oliynyk, Bogdana Solomko, Viktoriia Combinatorics Rings and Algebras 05C25, 05E16, 15B33, 16S50 The unitary Cayley graph $C_R$ of a finite unital ring $R$ is the simple graph with vertex set $R$ in which two elements $x$ and $y$ are connected by an edge if and only if $x-y$ is a unit of $R$. We characterize the unitary Cayley graph $C_{T_n (\mathbb{F})}$ of the ring of all upper triangular matrices $T_n(\mathbb{F})$ over a finite field $\mathbb{F}$. We show that $C_{T_n (\mathbb{F})}$ is isomorphic to the semistrong product of the complete graph $K_m$ and the antipodal graph of the Hamming graph $A(H(n,p^k))$, where $m=p^{\frac{kn(n-1)}{2}}$ and $|\mathbb{F}|=p^k$. In particular, if $|\mathbb{F}|=2$, then the graph $C_{T_n (\mathbb{F})}$ has $2^{n-1}$ connected components, each component is isomorphic to the complete bipartite graph $K_{m,m}$, where $m=2^{\frac{n(n-1)}{2}}$. We also compute the diameter, triameter, and clique number of the graph $C_{T_n (\mathbb{F})}$. |
| title | The unitary Cayley graph of upper triangular matrix rings |
| topic | Combinatorics Rings and Algebras 05C25, 05E16, 15B33, 16S50 |
| url | https://arxiv.org/abs/2403.01303 |