Symmetry And Metric Dimension Of Graphs

In the study of groups and graphs,the symmetry of a graph is a very active research topic.Symmetric graphs,especially small degree symmetric graphs,are often used to design interconnection network.Graphs associated with groups are related to automata theory and have appeared in various applications,as diverse as construction of larger networks,design of interconnection networks,data mining and combinatorial optimization.This thesis mainly studies the classification of cubic symmetric graphs and some properties of enhanced power graphs,reduced power graphs,order supergraph,commuting graphs and order divisor graph of finite groups,the main results of this thesis are as follows:We classify cubic symmetric graphs of order 88p~iand 22p~3,whereis a prime number and4)=1,2.We characterize the strong metric dimension of the order supergraph,the enhanced power graph and the reduced power graph of a finite group.As appli-cations,we compute the strong metric dimension of the above three classes of graphs for cyclic groups,dihedral groups and generalized quaternion groups.We characterize the strong metric dimension of the commuting graph of a finite group and give upper and lower bounds for the metric dimension of the commuting graph of a finite group.As applications,we compute the metric and strong metric dimension of the commuting graph of a dihedral group,a generalized quaternion group and a semidihedral group.We classify all finite groups whose order divisor graphs are dominatable and planar,and characterize all finite groups whose proper order divisor graphs admit a perfect code.As applications,we determine all abelian groups,dihedral groups and generalized quaternion groups whose proper order divisor graphs admit a total perfect code.