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Diagram semigroups are interesting algebraic and combinatorial objects, several types of them originating from questions in computer science and in physics. Here we describe diagram semigroups in a general framework and extend our…
A countable group $G$ is said to be \emph{matricial field} (MF) if it admits a strongly converging sequence of approximate homomorphisms into matrices; i.e, the norms of polynomials converge to those in the left regular representation. $G$…
A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the vertices of a stable set $V_i$ have the same degree $r_i$ for $i=1,2$. In this paper, we give an improved new Moore bound for an infinite family of such graphs with…
A finite group is called $\psi$-divisible iff $\psi(H)|\psi(G)$ for any subgroup $H$ of a finite group $G$. Here, $\psi(G)$ is the sum of element orders of $G$. For now, the only known examples of such groups are the cyclic ones of…
We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third…
Let $p$ be an odd prime. From a simple undirected graph $G$, through the classical procedures of Baer (Trans. Am. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lov\'asz (B. Braz. Math. Soc., 1989), there is a $p$-group $P_G$ of…
We present a structural approach of some results about jumps in the behavior of the profile (alias generating function) of hereditary classes of finite structures. We consider the following notion due to N.Thi\'ery and the second author. A…
Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…
We develop a battery of tools for studying quasi-isometric rigidity and classification problems for splittings of groups. The techniques work best for finite graphs of groups where all edge and vertex groups are coarse PD groups. For…
A finite group R is a CI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism x of R with S^x=T. The classification of CI-groups is an open problem in the theory…
Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the…
For a fixed finite set of finite tournaments ${\mathcal F}$, the ${\mathcal F}$-free orientation problem asks whether a given finite undirected graph $G$ has an $\mathcal F$-free orientation, i.e., whether the edges of $G$ can be oriented…
An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich…
A semi-projective representation is a homomorphism of a finite group into the group of semi-projective transformations of a finite dimensional vector space over a field. Schur's concept of a representation group for projective…
The first part of the paper centers in the study of embeddability between partially commutative groups. In [KK], for a finite simplicial graph $\Gamma$, the authors introduce an infinite, locally infinite graph $\Gamma^e$, called the…
In \cite{Oh22}, the second author defined a complex of groups decomposition of the fundamental group of a finitely generated 2-dimensional special group, called an \emph{intersection complex}, which is a quasi-isometry invariant. In this…
Let $R$ be a ring (not necessary commutative) with non-zero identity. The unit graph of $R$, denoted by $G(R)$, is a graph with elements of $R$ as its vertices and two distinct vertices $a$ and $b$ are adjacent if and only if $a+b$ is a…
The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in…
A semi-proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ with a weight function $w: A(D)\rightarrow \mathbb{Z}_+$, such that the in-weight of any adjacent vertices are distinct, where the in-weight of $v$ in…
In this paper, we determine bipartite graphs and complete graphs with horns, which are realizable as zero-divisor graphs of po-semirings. As applications, we classify commutative rings $R$ whose annihilating-ideal graph $\mathbb {AG}(R)$…