Related papers: When does an infinite ring have a finite compresse…
Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…
Let $R$ be a ring with unity. The graph $\Gamma(R)$ is a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. Let $\Gamma_2(R)$ is the subgraph of $\Gamma(R)$ induced by the…
In this paper, we investigate the existence of quantum fractional revival in unitary Cayley graphs over finite commutative rings with identity. We characterize all finite local rings that permit quantum fractional revival in their unitary…
The fundamental quandle is a complete invariant for unoriented tame knots \cite{JO, Ma} and non-split links \cite{FR}. The proof involves proving a relationship between the components of the fundamental quandle and the cosets of the…
Let R be an Artinian ring and G be the compressed zero-divisor graph associated to R. The question of when the clique number of compressed zero-divisor graphs is finite was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S.…
In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to…
Given a countable graph, we say a set $A$ of its vertices is \emph{universal} if it contains every countable graph as an induced subgraph, and $A$ is \emph{weakly universal} if it contains every finite graph as an induced subgraph. We show…
The commuting graph of a finite soluble group with trivial centre is investigated. It is shown that the diameter of such a graph is at most 8 or the graph is disconnected. Examples of soluble groups with trivial centre and commuting graph…
In finite graphs, finite-order tangles offer an abstract description of highly connected substructures. In infinite graphs, infinite-order tangles compactify the graphs in the same way the ends compactify connected locally finite graphs.…
We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an…
We show that there are four infinite prime graphs such that every infinite prime graph with no infinite clique embeds one of these graphs. We derive a similar result for infinite prime posets with no infinite chain or no infinite antichain.
Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$ is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x…
The commuting graph of a semigroup is the set of non-central elements; the edges are defined as pairs $(u,v)$ satisfying $uv=vu$. We provide an example of a field $F$ and an integer $n$ such that the commuting graph of…
Halin proved that every graph with an end $\omega$ containing infinitely many pairwise disjoint rays admits a subdivision of the infinite quarter-grid as a subgraph where all rays from that subgraph belong to $\omega$. We will prove a…
In this article, we introduce a new graph theoretic structure associated with a finite commutative ring, called nil clean divisor graph. For a ring $R$, nil clean divisor graph is denoted by $G_N(R)$, where the vertex set is $\{x\in R\,:\,…
We introduce and study asymptotically rigid mapping class groups of certain infinite graphs. We determine their finiteness properties and show that these depend on the number of ends of the underlying graph. In a special case where the…
The annihilator graph $AG(R)$ of the commutative ring $R$ is an undirected graph with vertex set as the set of all non-zero zero divisors of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann(xy) \neq ann(x) \cup…
We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in…
We will determine all infinite $2$-locally finite groups as well as infinite $2$-groups with planar subgroup graph and show that infinite groups satisfying the chain conditions containing an involution do not have planar embeddings. Also,…
We say that a graph is intrinsically non-trivial if every spatial embedding of the graph contains a non-trivial spatial subgraph. We prove that an intrinsically non-trivial graph is intrinsically linked, namely every spatial embedding of…