On groups all of whose Haar graphs are Cayley graphs

We construct and analyze an explicit basis for the homology of the boolean complex of a finite simple graph. This provides a combinatorial description of the spheres in the wedge sum representing the homotopy type of the complex. We assign a set of derangements to any finite simple graph. For each derangement, we construct a corresponding element in the homology of the complex, and the collection of these elements forms a basis for the homology of the boolean complex. In this manner, the spheres in the wedge sum describing the homotopy type of the complex can be represented by a set of derangements. We give an explicit, closed-form description of the derangements that can be obtained from any finite simple graph, and compute this set for several families of graphs. In the cases of complete graphs and Ferrers graphs, these calculations give bijective proofs of previously obtained enumerative results.

In this dissertation, we prove that if the flag complex on a finite simplicial graph is a 2-dimensional triangulated disk, then the Dehn function of the associated Bestvina--Brady group depends on the maximal dimension of the simplices in the interior of the flag complex. We also give some examples where the flag complex on a finite simplicial graph is not 2-dimensional, and we establish a lower bound for the Dehn function of the associated Bestvina--Brady group.

Symmetric edge polytopes $\mathcal{A}_G$ of type A are lattice polytopes arising from the root system $A_n$ and finite simple graphs $G$. There is a connection between $\mathcal{A}_G$ and the Kuramoto synchronization model in physics. In particular, the normalized volume of $\mathcal {A}_G$ plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph $G$, we give a formula for the $h^*$-polynomial of $\mathcal{A}_{\widehat{G}}$ by using matching generating polynomials, where $\widehat{G}$ is the suspension of $G$. This gives also a formula for the normalized volume of $\mathcal{A}_{\widehat{G}}$. Moreover, via the chemical graph theory, we show that for any cactus graph $G$, the $h^*$-polynomial of $\mathcal{A}_{\widehat{G}}$ is real-rooted. Finally, we extend the discussion to symmetric edge polytopes of type $B$, which are lattice polytopes arising from the root system $B_n$ and finite simple graphs.

Let Γ be a finite simple graph. If for some integer Γ is a Kn -free graph whose complement has an odd cycle of length at least then we say that Γ is an n-exact graph. For a finite group G, let denote the character graph built on the set of degrees of the irreducible complex characters of G. In this paper, we prove that the order of an n-exact character graph is at most Also we determine the structure of all finite groups G with extremal n-exact character graph.

Optimal identifying codes of two families of Cayley graphs

It is known that the polyomino ideal arising from a simple polyomino comes from a finite bipartite graph and, in particular, it is a prime ideal. A class of nonsimple polyominoes $\mathcal{P}$ for which the polyomino ideal $I_{\mathcal{P}}$ is a prime ideal and for which $I_{\mathcal{P}}$ cannot come from a finite simple graph will be presented.

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups. In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

Isomorphisms of Cayley multigraphs of degree 4 on finite abelian groups

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 extension graph of $\Gamma$. They show that each finite induced subgraph $\Delta$ of $\Gamma^e$ gives rise to an embedding between the partially commutative groups $G(\Delta)$ and $G({\Gamma})$. Furthermore, it is proven that in many instances the converse also holds. Our first result is the decidability of the Extension Graph Embedding Problem: there is an algorithm that given two finite simplicial graphs {\Delta} and {\Gamma} decides whether or not $\Delta$ is an induced subgraph of $\Gamma^e$. As a corollary we obtain the decidability of the Embedding Problem for 2-dimensional partially commutative groups. In the second part of the paper, we relate the Embedding Problem between partially commutative groups to the model-theoretic question of classification up to universal equivalence. We use our characterisation to transfer algebraic and algorithmic results on embeddability to model-theoretic ones and obtain some rigidity results on the elementary theory of atomic pc groups as well as to deduce the existence of an algorithm to decide if an arbitrary pc group is universally equivalent to a 2-dimensional one.

For [Formula: see text], a [Formula: see text]-firecracker graph, denoted by [Formula: see text], is obtained by concatenation of [Formula: see text] [Formula: see text]-star graphs by connecting one leaf from each [Formula: see text]-star graph. Given a finite simple graph [Formula: see text], one can associate a simplicial complex [Formula: see text]. In this paper, we compute all the graded Betti numbers of the edge ideal [Formula: see text] of the firecracker graph [Formula: see text] by using the combinatorial data associated with the simplicial complex [Formula: see text] We also find the regularity of the ideals [Formula: see text]. Further, using the domination parameters of the graphs, we explicitly compute the projective dimension of [Formula: see text].

An n-crown on 2n vertices is a graph obtained from complete bipartite graph by removing edges of a perfect matching. Given a finite simple graph G, one can associate a simplicial complex . In this article, we use combinatorial data from the associated simplicial complex of the crown graph and give formulas to find all Betti numbers of edge ideals of .

Many popular interconnection network topologies, such as hypercubes and toroidal meshes, are based on Cayley graphs of Abelian groups. The symmetry and algebraic structure of these graphs result in many nice physical properties of the network concerning layout, routing algorithms, and load balancing. There has been interest in low-diameter Abelian--Cayley graphs because of their smaller communication delay and reduced congestion. For any fixed number of nodes n, and any fixed out-degree k, we are interested in how small the diameter of directed Cayley graphs of Abelian groups can be and what these low-diameter graphs look like. We give an upper bound of $n \leq \frac{3(d + 3)^3}{25}$ for the size of directed Abelian--Cayley graphs with k = 3 and diameter d, correcting a previously published result by Hsu and Jia [SIAM J. Discrete Math., 7 (1994), pp. 57--71]. Our method is based on translational tiling techniques and is a generalization of Wong and Coppersmith's method for k = 2 [J. Assoc. Comput. Mach., 21 (1974), pp. 392--402]. Moreover, our method works for all Abelian groups, not just the cyclic case. For k = 3 we give computational results for the largest Abelian--Cayley graph as a function of diameter. When n = 84 m3 , for integer m, there is a network with $n = \frac{(d + 3)^3}{11.95}$ whose diameter is approximately three-fourths of that of a three-dimensional toroidal cube.

Given a finite simplicial graph ${\cal G}$, the graph group $G{\cal is the group with generators in one-to-one correspondence with the vertices of ${\cal G}$ and with relations stating two generators commute if their associated vertices are adjacent in ${\cal G}$. The Bieri-Neumann-Strebel invariant can be explicitly described in terms of the original graph ${\cal G}$ and hence there is an explicit description of the distribution of finitely generated normal subgroups of $G{\cal G}$ with abelian quotient. We construct Eilenberg-MacLane spaces for graph groups and find partial extensions of this work to the higher dimensional invariants.

Starting from any finite simple graph, one can build a reflexive polytope known as a symmetric edge polytope. The first goal of this paper is to show that symmetric edge polytopes are intrinsically matroidal objects: more precisely, we prove that two symmetric edge polytopes are unimodularly equivalent precisely when they share the same graphical matroid. The second goal is to show that one can construct a generalized symmetric edge polytope starting from every regular matroid. Just like in the usual case, we are able to find combinatorial ways to describe the facets and an explicit regular unimodular triangulation of any such polytope. Finally, we show that the Ehrhart theory of the polar of a given generalized symmetric edge polytope is tightly linked to the structure of the lattice of flows of the dual regular matroid.

A finite simple graph G = (V, E) admits an H-covering if every edge of G belongs to a subgraph of G isomorphic to H. We said G to be an H-magic if there exists a total labeling f: V (G) ∪ E(G) → {1,2,…, |V(G)| + |E(G)|}, such that for each subgraph H′ = (V′, E′) of G isomorphic to H, the magic sum m (f) = Σv∈V′ f(v) + Σe∈E′ f(e). Then, G is H-supermagic if f(V) = {1,2,…,|V|}. This research provides star-supermagicness on helm graph and Sn ⊙ K¯m.