Commuting graph of some prime order elements in symplectic and mathieu groups

This thesis concentrates on an investigation to describe the properties of groups in graph-theoretic context such as connectivity, distance and many more. One of the ways to achieve this is by studying a commuting graph. Let G be a group and X is a subset of G. The commuting graph C (G,X) is defined by taking a vertex set X, and letting two distinct vertices x, y ∈ X be adjacent whenever xy = yx. We construct the commuting graph by selecting G as a symplectic group or a Mathieu group, and X denotes a conjugacy class of G for prime order elements. The main contribution of this thesis deals with various aspects of the structure of the commuting graph C (G,X). We choose t ∈ X to be a fixed vertex of C (G,X), then we continue to determine the connectivity of the graph whether it is connected or disconnected. If C (G,X) is found to be connected, then we further the analysis in the commuting graph C (G,X) to obtain the distance of x ∈ X from t, the disc size and its diameter. However, if the commuting graph C (G,X) is disconnected, then there are more than one subgraph can be obtained. These subgraphs which are named as subgraphs D (G,X) of C (G,X) are then found to be isomorphic with each other and the diameter of D (G,X) is either 1 or 2 for the cases we consider. Consequently the number of the subgraphs can also be determined and for each subgraph D (G,X), we describe its structure as we treat the connected commuting graph C (G,X). Next, the study of the commuting graph takes naturally into consideration of the suborbits of G on X, that is the orbits under the action of a centralizer CG(t). Apart from finding the sizes of CG(t)-orbits of X, we obtain representatives x ∈ X for each of these orbits. Moreover, we also include some properties which in many cases aid speedy identification of the given orbit. We obtain the subgroup ⟨t, x⟩ generated by t and x. Then, we conduct a deeper analysis on the CG(t)-orbits by specifying the connectedness between every two orbits. We visualize the interaction of the CG(t)-orbits in a collapsed adijacency diagram, showing a line to join two circles of so-called CG(t)-orbits. We also compute how many vertices a CG(t)-orbit representative is connected to and record those data in a collapsed adjacency matrix. The entries of that matrix may indicate whether two distinct orbits are adjacent or not in the graph. Finally, we present the spectrum of the collapsed adjacency matrix of the commuting graph, that is the multiset of eigenvalues of its collapsed adjacency matrix and of course is one of many important invariants from which much information about the graph can be ascertained. The study of the spectrum of the graph, Spec(G,X) relates to the graph energy, ε(G,X). We obtain ε(G,X) > 0 if the graph is connected, otherwise ε(G,X) = 0.

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