arXiv subjects
Publications and source records attributed to Daniel G. Mendoza.
The non--commuting graph $Γ(G)$ of a non--abelian group $G$ is defined as follows. The vertex set $V(Γ(G))$ of $Γ(G)$ is $G\setminus Z(G)$ where $Z(G)$ denotes the center of $G$ and two vertices $x$ and $y$ are adjacent if and only if $xy\neq yx$. For non--abelian finite groups $G$ and $H$ it is conjectured that if $Γ(G) \cong Γ(H)$, then $|G|=|H|$. We prove the conjecture.
Let $G$ be a finite group, the enhanced power graph of $G$, denoted by $Γ_G^e$, is the graph with vertex set $G$ and two vertices $x,y$ are edge connected in $Γ_{G}^e$ if there exist $z\in G$ such that $x,y\in\langle z\rangle$. Let $ζ$ be a edge-coloring of $Γ_G^e$. In this article, we calculate the rainbow connection number of the enhanced power graph $Γ_G^e$.
We introduce the combinatorial Lyubeznik resolution of monomial ideals. We prove that this resolution is isomorphic to the usual Lyubezbnik resolution. As an application, we give a combinatorial method to determine if an ideal is a Lyubeznik ideal. Furthermore, the minimality of the Lyubeznik resolution is characterized and we classify all the Lyubeznik symbols using combinatorial criteria. We get a combinatorial expression for the projective dimension, the length of Lyubeznik, and the arithmetical rank of a monomial ideal. We define the Lyubeznik totally ideals as those ideals that yield a minimal free resolution under any total order. Finally, we present that for a family of graphics, that their edge ideals are Lyubeznik totally ideals.
We give upper bounds for the Stanley depth of edge ideals of certain k-partite clutters. In particular, we generalize a result of Ishaq about the Stanley depth of the edge ideal of a complete bipartite graph. A result of Pournaki, Seyed Fakhari and Yassemi implies that the Stanley's conjecture holds for d-uniform complete d-partite clutters. Here we give a shorter and different proof of this fact.
For a finite group $G$ with a normal subgroup $H$, the enhanced quotient graph of $G/H$, denoted by $\mathcal{G}_{H}(G),$ is the graph with vertex set $V=(G\backslash H)\cup \{e\}$ and two vertices $x$ and $y$ are edge connected if $xH = yH$ or $xH,yH\in \langle zH\rangle$ for some $z\in G$. In this article, we characterize the enhanced quotient graph of $G/H$. The graph $\mathcal{G}_{H}(G)$ is complete if and only if $G/H$ is cyclic, and $\mathcal{G}_{H}(G)$ is Eulerian if and only if $|G/H|$ is odd. We show some relation between the graph $\mathcal{G}_{H}(G)$ and the enhanced power graph $\mathcal{G}(G/H)$ that was introduced by Sudip Bera and A.K. Bhuniya (2016). The graph $\mathcal{G}_H(G)$ is complete if and only if $G/H$ is cyclic if and only if $\mathcal{G}(G/H)$ is complete. The graph $\mathcal{G}_H(G)$ is Eulerian if and only if $|G|$ is odd if and only if $\mathcal{G}(G)$ is Eulerian, i.e., the property of being Eulerian does not depend on the normal subgroup $H$.