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 defi...
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Format: | Thesis |
Language: | English |
Published: |
2020
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Subjects: | |
Online Access: | http://psasir.upm.edu.my/id/eprint/90776/1/IPM%202020%208%20%20-%20IR.pdf |
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Summary: | 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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