Narrow Results

If $R$ is a ring then the square element graph $𝕊q(R)$ is the simple undirected graph whose vertex set consists of all non-zero elements of $R$ and two distinct vertices $u,v$ are adjacent if and only if $u+v=x^2$ for some $x\in R\setminus \{0\}$. In this paper, we provide some necessary and sufficient conditions for the connectedness of $𝕊q(R)$, where $R$ is a ring with identity. We mainly characterize some special class of ring $R$ which we call square-subtract ring for which the graph $𝕊q(M_n(R))$ is connected.

Let $R$ be a commutative ring with identity, $n\ge 2$ be a positive integer and $M_n(R)$ be the set of all $n\times n$ matrices over $R.$ For a matrix $A\in M_n(R),$ Tr$(A)$ is the trace of $A.$ The trace graph of the matrix ring $M_n(R),$ denoted by ${\mathrm{\Gamma }}_t(M_n(R)),$ is the simple undirected graph with vertex set $\{A\in M_n{(R)}^{\ast }:\text{there exists }B\in M_n{(R)}^{\ast }$$\text{ such that Tr}(AB)=0\}$ and two distinct vertices $A$ and $B$ are adjacent if and only if Tr$(AB)=0.$ The ideal-based trace graph of the matrix ring $M_n(R)$ with respect to an ideal $I$ of $R,$ denoted by ${\mathrm{\Gamma }}_{I^t}(M_n(R)),$ is the simple undirected graph with vertex set $M_n(R)\setminus M_n(I)$ and two distinct vertices $A$ and $B$ are adjacent if and only if Tr$(AB)\in I.$ In this paper, we investigate some properties and structure of ${\mathrm{\Gamma }}_{I^t}(M_n(R)).$ Further, it is proved that both ${\mathrm{\Gamma }}_t(M_n(R))$ and ${\mathrm{\Gamma }}_{I^t}(M_n(R))$ are Hamiltonian.