Narrow Results

The orthogonality graph$O(R)$ of a ring $R$ is the undirected graph with vertex set consisting of all nonzero two-sided zero divisors of $R$, in which for two vertices $x$ and $y$ (needless distinct), $x\sim y$ is an edge if and only if $xy=yx=0$. Let $n\ge 2$, ${\mathrm{\text{Mat}}}_n(F)$ be the set of all $n\times n$ matrices over a finite field $F$, and $R_n(F)$ the subset of ${\mathrm{\text{Mat}}}_n(F)$ consisting of all rank one upper triangular matrices. In this paper, we describe the full automorphism group, and using the technique of generalized equivalent canonical form of matrices, we compute the fixing number of $O(R_n(F))$, the induced subgraph of $O({\mathrm{\text{Mat}}}_n(F))$ with vertex set $R_n(F)$.

In this paper, we consider the automorphisms of the strong semilattice of groups and relate them to the isomorphisms and automorphisms of underlying groups. We also provide a construction for non-trivial automorphisms of semilattices.

In this paper, we give an explicit expression for the number of pairwise non-isomorphic connected regular graphs as well as odd graphs on n vertices, where $n\ge 2$ is a positive integer.