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M. K. Sen

Publications and source records attributed to M. K. Sen.

5 recordsLinked to original sources
arXiv2026

Generalized Graph Compositions with Applications to Difference Graphs of Finite Groups

The difference graph $D(G)$ of a finite group $G$ is obtained from the edge difference between its intersection power graph and power graph, after deleting isolated vertices. This graph has already been studied, with sufficient conditions for connectedness and a diameter bound $6$ for finite groups satisfying those conditions. We use generalized graph composition to reduce $D(G)$ to a graph $B(G)$ on the cyclic subgroups of $G$, so that connectedness and diameter are determined by the subgroup structure of $G$. We obtain a general criterion for the non-emptiness of $B(G)$ in terms of branching subgroups and b-normality, and characterize its connectedness for finite $p$-groups, non-cyclic finite abelian groups, and non-abelian groups with both trivial and non-trivial center. Combined with the previously established cyclic-group case, this gives a complete characterization of non-emptiness and connectedness of difference graphs for all finite groups. The successive structural cases lead naturally to the sharp diameter bounds $2,3,4,$ and $5$. For centerless non-abelian groups, connectedness is governed either by a unique branching subgroup or by an auxiliary graph $\mathcal A(G)$; in the latter case \[ \operatorname{diam}\mathcal A(G)-1 \leq \operatorname{diam}B(G) \leq \max\{4,\operatorname{diam}\mathcal A(G)+1\}, \] and both bounds are sharp.

math.GR↗
arXiv2020

The endomorphism semiring of a commutative inverse semigroup

The authors [3] proved that the endomorphism semiring of a nontrivial semilattice is always subdirectly irreducible and described its monolith. Here we prove that the endomorphism semiring of a commutative inverse semigroup with at least two idempotents is always subdirectly irreducible and describe its monolith.

math.RA↗
arXiv2017

On the idempotent semirings such that $\mathcal{D}^\bullet$ is the least distributive lattice congruence

Here we describe the least distributive lattice congruence $η$ on an idempotent semiring in general and characterize the varieties $D^\bullet, L^\bullet$ and $R^\bullet$ of all idempotent semirings such that $η=\mathcal{D}^\bullet, \mathcal{L}^\bullet$ and $\mathcal{R}^\bullet$, respectively. If $S \in D^\bullet [L^\bullet, R^\bullet]$, then the multiplicative reduct $(S, \cdot)$ is a [left, right] normal band. Every semiring $S \in D^\bullet$ is a spined product of a semiring in $L^\bullet$ and a semiring in $R^\bullet$ with respect to a distributive lattice.

math.RA↗
arXiv2016

An introduction to coding sequences of graphs

In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over $\mathbb{Z}_2$ which has the consecutive $1$'s property (i.e., $1$'s are consecutive) for columns and obtained a characterization of graphic matroids in terms of this. In fact, we introduce a new representation of simple undirected graphs in terms of some vectors of finite dimensional vector spaces over $\mathbb{Z}_2$ which satisfy consecutive $1$'s property. The set of such vectors is called a coding sequence of a graph $G$. Among all such coding sequences we identify the one which is unique for a class of isomorphic graphs. We call it the code of the graph. We characterize several classes of graphs in terms of coding sequences. It is shown that a graph $G$ with $n$ vertices is a tree if and only if any coding sequence of $G$ is a basis of the vector space $\mathbb{Z}_2^{n-1}$ over $\mathbb{Z}_2$. Moreover considering coding sequences as binary matroids, we obtain a characterization for simple graphic matroids and found a necessary and sufficient condition for graph isomorphism in terms of a special matroid isomorphism between their corresponding coding sequences. For this, we introduce the concept of strong isomorphisms of segment binary matroids and show that two simple (undirected) graphs are isomorphic if and only if their canonical sequences are strongly isomorphic segment binary matroids.

math.CO↗