General form of domination polynomial for two types of graphs associated to dihedral groups

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups

Commuting involution graphs for [(A)\tilde]n

In this article we consider the commuting graphs of involution conjugacy classes in the affine Weyl group A~n. We show that where the graph is connected the diameter is at most 6. MSC(2000): 20F55, 05C25, 20D60

Embedability between right-angled Artin groups

In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph \gam, we produce a new graph through a purely combinatorial procedure, and call it the extension graph \gam^e of \gam. We produce a second graph \gam^e_k, the clique graph of \gam^e, by adding extra vertices for each complete subgraph of \gam^e. We prove that each finite induced subgraph $\Lambda$ of \gam^e gives rise to an inclusion A(\Lambda)\to A(\gam). Conversely, we show that if there is an inclusion A(\Lambda)\to A(\gam) then $\Lambda$ is an induced subgraph of \gam^e_k. These results have a number of corollaries. Let $P_4$ denote the path on four vertices and let $C_n$ denote the cycle of length $n$. We prove that $A(P_4)$ embeds in A(\gam) if and only if $P_4$ is an induced subgraph of \gam. We prove that if $F$ is any finite forest then $A(F)$ embeds in $A(P_4)$. We recover the first author's result on co--contraction of graphs and prove that if \gam has no triangles and A(\gam) contains a copy of $A(C_n)$ for some $n\geq 5$, then \gam contains a copy of $C_m$ for some $5\le m\le n$. We also recover Kambites' Theorem, which asserts that if $A(C_4)$ embeds in A(\gam) then \gam contains an induced square. Finally, we determine precisely when there is an inclusion $A(C_m)\to A(C_n)$ and show that there is no "universal" two--dimensional right-angled Artin group.Comment: 35 pages. Added an appendix and a proof that the extension graph is quasi-isometric to a tre

The method of the weakly conjugate operator: Extensions and applications to operators on graphs and groups

In this review we present some recent extensions of the method of the weakly conjugate operator. We illustrate these developments through examples of operators on graphs and groups.Comment: 11 page