The unitary Cayley graph of upper triangular matrix rings

2024/03/02 by Waldemar Hołubowski, Hołubowski, Waldemar, Sergiy Kozerenko +5
Engineering · Mathematics · #05C25 #05E16 #15B33 #16S50 #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2403.01303

openalex publication_date 2024/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 (\mathbbF) of the ring of all upper triangular matrices T_n(\mathbbF) over a finite field \mathbbF. We show that C_T_n (\mathbbF) 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^{(kn(n-1))/(2)} and |\mathbbF|=p^k. In particular, if |\mathbbF|=2, then the graph C_T_n (\mathbbF) has 2^n-1 connected components, each component is isomorphic to the complete bipartite graph K_m,m, where m=2^(n(n-1))/(2). We also compute the diameter, triameter, and clique number of the graph C_T_n (\mathbbF).

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