Narrow Results

For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors. Both results also include suitable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results.

We find a set of generators for the automorphism group Aut G of a graph product G of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup Aut^⋆ G defined in [Automorphisms of graph products of abelian groups, to appear in Groups, Geometry and Dynamics]. We follow closely the plan of M. Laurence's paper [A generating set for the automorphism group of a graph group, J. London Math. Soc. (2)52(2) (1995) 318–334].

It is shown that the compressed word problem of a graph product of finitely generated groups is polynomial time Turing-reducible to the compressed word problems of the vertex groups. A direct corollary of this result is that the word problem for the automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Moreover, it is shown that a restricted variant of the simultaneous compressed conjugacy problem is polynomial time Turing-reducible to the same problem for the vertex groups. A direct corollary of this result is that the word problem for the outer automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Finally, it is shown that the compressed variant of the ordinary conjugacy problem can be solved in polynomial time for right-angled Artin groups.

In this paper, we generalize Magnus' Freiheitssatz and solution to the word problem for one-relator groups by considering one-relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups.

We study properties of automorphisms of graph products of groups. We show that graph product $\mathrm{\Gamma }{𝒢}$ has nontrivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has nontrivial pointwise inner automorphisms. We use this result to study residual finiteness of $\mathrm{\text{Out}}(\mathrm{\Gamma }{𝒢})$. We show that if all vertex groups are finitely generated residually finite and the vertex groups corresponding to central vertices satisfy certain technical (yet natural) condition, then $\mathrm{\text{Out}}(\mathrm{\Gamma }{𝒢})$ is residually finite. Finally, we generalize this result to graph products of residually $p$-finite groups to show that if $\mathrm{\Gamma }{𝒢}$ is a graph product of finitely generated residually $p$-finite groups such that the vertex groups corresponding to central vertices satisfy the $p$-version of the technical condition then $\mathrm{\text{Out}}(\mathrm{\Gamma }{𝒢})$ is virtually residually $p$-finite. We use this result to prove bi-orderability of Torreli groups of some graph products of finitely generated residually torsion-free nilpotent groups.

We show that there exists no left order on the free product of two nontrivial, finitely generated, left-orderable groups such that the corresponding positive cone is represented by a regular language. Since there are orders on free groups of rank at least two with positive cone languages that are context-free (in fact, 1-counter languages), our result provides a bound on the language complexity of positive cones in free products that is the best possible within the Chomsky hierarchy. It also provides a strengthening of a result by Cristóbal Rivas which states that the positive cone in a free product of nontrivial, finitely generated, left-orderable groups cannot be finitely generated as a semigroup. As another illustration of our method, we show that the language of all geodesics (with respect to the natural generating set) that represent positive elements in a graph product of groups defined by a graph of diameter at least 3 cannot be regular.

The notion of a generalized scale emerged in recent joint work with Afsar–Brownlowe–Larsen on equilibrium states on $C^{\ast }$-algebras of right Least Common Multiple (LCM) monoids, where it features as the key datum for the dynamics under investigation. This work provides the structure theory for such monoidal homomorphisms. We establish the uniqueness of the generalized scale and characterize its existence in terms of a simplicial graph arising from a new notion of irreducibility inside right LCM monoids. In addition, the method yields an explicit construction of the generalized scale if existent. We discuss applications for graph products as well as algebraic dynamical systems and reveal a striking connection to Saito’s degree map.

We prove that the Connes embedding problem is stable under graph products.

We prove that the rapid decay property (RD) of groups is preserved by graph products defined on finite simplicial graphs.

The concept of zero-divisor graph of a commutative ring was introduced by Beck [Coloring of commutating ring, J. Algebra116 (1988) 208–226]. In this paper, we present some properties of zero divisor graphs obtained from ring $Z_p\times Z_q\times Z_r$, where $p,q$ and $r$ are primes. Also, we give some degree-based topological indices of this special graph.

In this paper, we introduce a graph product, namely, $M$-bicone product and determine the generalized characteristic polynomial of the graphs obtained from this product. Consequently, we obtain the characteristic polynomial of the adjacency matrix, the Laplacian matrix and the signless Laplacian matrix of this graph. Also, from these results, we obtain the $L$-spectra of some families of bicone product of graphs. As an application, we obtain infinite pairs of $L$-cospectral graphs.

This paper studies the 2-distance chromatic number of some graph product. A coloring of $G$ is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of $G$ is the 2-distance chromatic number and denoted by ${\chi }_2(G)$. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

A star coloring of a graph $G$ is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors have connected components that are star graphs. A graph $G$ is $k$- star-colorable if there exists a star coloring of $G$ from a set of $k$ colors. The minimum positive integer $k$ for which $G$ is $k$-star-colorable is the star chromatic number of $G$ and is denoted by ${\chi }_s(G)$. In this paper, upper and lower bounds are presented for the star chromatic number of the rooted product, hierarchical product, and lexicographic product.