arXiv subjects
Publications and source records attributed to Shahram Mehry.
We introduce the \emph{intersection orbital graph} $\Gamma(G_1, G_2; \Omega)$ associated with two permutation groups $G_1, G_2 \leq \mathrm{Sym}(\Omega)$ on a finite set $\Omega$.
We prove that the canonical orientation of the generalized Kneser graph $KG(n,k,s)$ contains a directed Hamiltonian cycle for all integers $s \geq 3$ and $n>sk$. Furthermore, we establish that the dichromatic number of this oriented graph is exactly $k$. As a special case, our results apply to the $s$-stable Kneser graphs $K_{s\text{-stab}}(n,k)$, resolving their directed Hamiltonicity and dichromatic number. Our proof adapts the class graph framework of Ledezma and Pastine to the directed setting, leveraging cyclic rotations and friend class adjacencies to construct a single directed cycle spanning all vertices. This work provides a unified and strengthened perspective on the Hamiltonian properties of Kneser-type graphs.
Let $M$ be a left $R$-module. We define the \emph{homomorphism submodule graph} $\Gamma_{\mathrm{Hom}}(M)$ as the simple graph whose vertices are the proper submodules of $M$, with an edge between distinct vertices $N_1$ and $N_2$ if and only if $\mathrm{Hom}_R(N_1, M/N_2) \ne 0$ or $\mathrm{Hom}_R(N_2, M/N_1) \ne 0$. This graph encodes homological information about $M$ and reflects its internal structure. We compute $\Gamma_{\mathrm{Hom}}(M)$ for semisimple and uniserial modules, establish precise correspondences between graph-theoretic and algebraic properties, and prove that for modules over Artinian local rings, the isomorphism type of $M$ is determined by $\Gamma_{\mathrm{Hom}}(M)$. We also show that over commutative rings with identity, the graph is always chordal, and we relate its spectral radius to composition length in natural families.
Let $R$ be a commutative ring with identity. We introduce a novel bipartite graph $\mathcal{B}(R)$, the \textit{bipartite zero-divisor--unit graph}, whose vertex set is the disjoint union of the nonzero zero-divisors $Z(R)^*$ and the unit group $U(R)$. A vertex $z \in Z(R)^*$ is adjacent to $u \in U(R)$ if and only if $z + u \in Z(R)$. This construction provides an \textit{additive} counterpart to the well-established \textit{multiplicative} zero-divisor graphs. We investigate fundamental graph-theoretic properties of $\mathcal{B}(R)$, including connectedness, diameter, girth, chromatic number, and planarity. Explicit descriptions are given for rings such as $\mathbb{Z}_n$, finite products of fields, and local rings. Our results are sharpest for \textit{finite reduced rings}, where $\mathcal{B}(R)$ yields a graphical characterization of fields and serves as a complete invariant: $\mathcal{B}(R) \cong \mathcal{B}(S)$ implies $R \cong S$ for finite reduced rings $R$ and $S$. The graph also reveals structural distinctions between reduced and non-reduced rings, underscoring its utility in the interplay between ring-theoretic and combinatorial properties.