The Unitary Cayley Graph of Upper Triangular Matrix Rings
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Date
2024
Authors
Hołubowski, Waldemar
Kozerenko, Sergiy
Oliynyk, Bogdana
Solomko, Viktoriia
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Abstract
The unitary Cayley graph CR 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 CTn(F) of the ring of all upper triangular matrices Tn(F) over a finite field F. We show that CTn(F) is isomorphic to the semistrong product of the complete graph Km and the antipodal graph of the Hamming graph A(H(n, pk)), where m = p kn(n−1) 2 and |F| = pk. In particular, if |F| = 2, then the graph CTn(F) has 2n−1 connected components, each component is isomorphic to the complete bipartite graph Km,m, where m = 2 n(n−1) 2 . We also compute the diameter, triameter, and clique number of the graph CTn(F).
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Keywords
unitary Cayley graph, upper triangular matrix ring, antipodal graph, Hamming graph, clique number, preprint
Citation
The Unitary Cayley Graph of Upper Triangular Matrix Rings / Waldemar Hołubowski, Sergiy Kozerenko, Bogdana Oliynyk, Viktoriia Solomko - [S. l.] : arXiv, 2024. - 9 p. - https://doi.org/10.48550/arXiv.2403.01303