Journal of Algebraic Structures and Their Applications
Volume:12 Issue: 1, Winter 2025

For a ring $R$, the intersection minimal ideal graph, denoted by $ \wedge(R) $, is a simple undirected graph whose vertices are proper non-zero (right) ideals of $R$ and any two distinct vertices $I_{1}$ and $I_{2}$ are adjacent if and only if $ I_{1} \cap I_{2}$ is a minimal ideal of $R$. In this article, we explore connectedness, clique number, split character, planarity, independence number, domination number of $\wedge(R)$.

We introduce the notions of commuting automorphism and commutator polygroups. The basic question that can be arose about the set of all commuting automorphisms is that for the assumed polygroup $ ( P, \cdot )$, under what conditions the set of all commuting automorphism $ \boldsymbol {A} ( P) $ is a subgroup of $ Aut ( P)$. In this paper basically the answer to this question is investigated for the class of polygroups.

In this paper we introduce the concepts of special regular clean elements and regular clean decomposition in a ring R. These concepts lead us to the notion of special regular clean ring. We prove that for a special regular clean element $a=e+r \in R$ and unit $u\in R$ then $au$ is a special regular clean if $u$ is an inner inverse of $e$. We establish that an abelian ring $R$ is a special regular clean ring if and only if the twisted power series ring $R[[x,\sigma]]$ is a special regular clean ring. We also study various characterizations of special clean and special regular clean rings.

In this paper we compute spectrum, Laplacian spectrum, signless Laplacian spectrum and their corresponding energies of commuting conjugacy class graph of the group $G(p, m, n) = \langle x, y : x^{p^m} = y^{p^n} = [x, y]^p = 1, [x, [x, y]] = [y, [x, y]] = 1\rangle$, where $p$ is any prime, $m \geq 1$ and $n \geq 1$. We derive some consequences along with the fact that commuting conjugacy class graph of $G(p, m, n)$ is super integral. We also compare various energies and determine whether commuting conjugacy class graph of $G(p, m, n)$ is hyperenergetic, L-hyperenergetic or Q-hyperenergetic.

Let $L$ be a lattice with $1$. The meet-nonessential graph $\mathbb{MG} (L)$ of $L$ is a graph whose vertices are all nonessential filters of $L$ and two distinct filters $F$ and $G$ are adjacent if and only if $F \wedge G$ is a nonessential filter of $L$. The basic properties and possible structures of the graph $\mathbb{MG}(L)$ are investigated. The clique number, domination number and independence number of $\mathbb{MG}(L)$ and their relations to algebraic properties of $L$ are explored.

Let $\mathcal{P}_q(n)$ be the set of all subspaces in the vector space $\mathbb{F}_q^n$. There is a subspace distance $d_S(U,V)$ between any two subspaces $U$ and $V$. A subspace code is also a subset of $\mathcal{P}_q(n)$. It is known that $d_S(U,V)\geq d_H(\nu(\pi U),\nu(\pi V))$, where $\pi\in S_n$, $\nu(U)$ denotes the pivot vector of $E(U)$ and $E(U)$ is the reduced row echelon form of the generator matrix of $U$. In this paper, we show that if $E(U)$ and $E(V)$ have at most one non-zero entry in each rows and each columns then the equality holds. Moreover, we introduce the sets $\mathcal{G}_{U,V}=\{\pi\in S_n\mid d_S(U,V)=d_H(\nu(\pi U),\nu(\pi V))\}$ for any $U,V\in\mathcal{P}_q(n)$ and examine them in the spaces $\mathcal{P}_2(4)$, $\mathcal{P}_2(5)$, $\mathcal{P}_2(6)$ and $\mathcal{P}_3(4)$. It is shown that the groups $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $S_4$ and $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $D_8$, $S_3\times \mathbb{Z}_2$, $S_4$, $S_5$ appears between these sets in $\mathcal{P}_2(4)$ and $\mathcal{P}_2(5)$, respectively. Moreover, the groups $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $D_8$, $\mathbb{Z}_2\times \mathbb{Z}_2 \times \mathbb{Z}_2$, $S_3\times \mathbb{Z}_2$, $D_8\times \mathbb{Z}_2$, $S_4$, $S_3\times S_3$, $S_4\times \mathbb{Z}_2$, $(S_3\times S_3)$:$2$, $S_5$, $S_6$ and $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $D_8$, $S_4$ appears between these sets in $\mathcal{P}_2(6)$ and $\mathcal{P}_3(4)$, respectively.