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To any simple graph \(G\), the clique graph operator \(K\) assigns the graph \(K(G)\) which is the intersection graph of the maximal complete subgraphs of \(G\). The iterated clique graphs are defined by \(K^{0}(G)=G\) and…

An orientation of a graph is semi-transitive if it contains no directed cycles and has no shortcuts. An undirected graph is semi-transitive if it can be oriented in a semi-transitive manner. The class of semi-transitive graphs includes…

The complete double vertex graph $M_2(G)$ of $G$ is defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent…

Given a definably compact group G in a saturated o-minimal structure, there is a canonical homomorphism from G to a compact real Lie group F(G). We establish a similar result for the (o-mininimal) universal cover of a definably compact…

A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisson manifold with a real symplectic manifold, but in possibly many different ways. In this paper we show that the isomorphism class of the…

An independent set in a graph is a set of pairwise non-adjacent vertices, and a(G) is the size of a maximum independent set in the graph G. If s_{k} is the number of independent sets of cardinality k in G, then…

The graph polynomial for the number of independent sets of size $k$ in a general undirected graph is shown to be equal to an elementary symmetric polynomial of the vertex monomials, which are determined by the edges incident at the…

We study graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the fractional clique number. We establish some basic properties of the independence set polytope in the…

In this paper, we introduce a graph structure called linear dependence graph of a finite dimensional vector space over a finite field. Some basic properties of the graph like connectedness, completeness, planarity, clique number, chromatic…

The collapsibility number of simplicial complexes was introduced by Wegner in order to understand the intersection patterns of convex sets. This number also plays an important role in a variety of Helly type results. We show that the…

Let $f,g:M \rightarrow N$ be two maps between simply-connected smooth manifolds $M$ and $N$, such that $M$ is compact and $N$ is of finite $\mathbb{R}$-type. The goal of this paper is to use integration of certain differential forms to…

In this paper we study the homotopy type of $\Hom(C_m,C_n)$, where $C_k$ is the cyclic graph with $k$ vertices. We enumerate connected components of $\Hom(C_m,C_n)$ and show that each such component is either homeomorphic to a point or…

The classic theorem of Gallai and Milgram (1960) generalizes several fundamental results in Graph Theory, such as Dilworth's theorem on posets and K\H{o}nig's theorem on matchings in bipartite graphs. The theorem asserts that for every…

We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast…

An independent set $I_c$ is a \textit{critical independent set} if $|I_c| - |N(I_c)| \geq |J| - |N(J)|$, for any independent set $J$. The \textit{critical independence number} of a graph is the cardinality of a maximum critical independent…

Let M be the cotangent bundle of S^2, with the standard symplectic structure. By adapting an argument of Gromov we determine the weak homotopy type of the group S of those symplectic automorphisms of M which are trivial at infinity. It…

A graph class $\mathcal C$ is monadically dependent if one cannot interpret all graphs in colored graphs from $\mathcal C$ using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized by…

We describe the second homotopy group of any CW-complex $K$ by analyzing the universal cover of a locally finite model of $K$ using the notion of $G$-coloring of a partially ordered set. As applications we prove a generalization of the…

For any given graph $G$ consider a graph $\widetilde{G}$ which is a cone over graph $G.$ In this paper, we study two important invariants of such a cone. Namely, complexity (the number of spanning trees) and the Jacobian of a graph. We…

Given two graphs $G$ and $H$, assume that $\mathscr{C}=\{C_1,C_2,\ldots, C_q\}$ is a clique cover of $G$ and $U$ is a subset of $V(H)$. We introduce a new graph operation called the clique cover product, denoted by $G^{\mathscr{C}}\star…