On metric dimension and determining number of certain zero divisor graphs

Abstract

This thesis mainly focuses on the element-wise and substructure-wise interplay newlinebetween graph theory and commutative algebra. The thesis contains two major topics newlinesuch as the zero-divisor graph and#915;(R) of a commutative ring R, and the group vertex newlinemagicness of graphs. newlineFirst, the structure of zero-divisor graph of semisimple group ring FqCn, where Fq newlineis a finite field with q elements and Cn is the cyclic group with n elements, is studied. newlineAlso, the order, size, degree, chromatic number, connectivity, and domination number newlineare obtained. In addition, the bipartiteness, regularity, planarity, perfectness, and newlineEulerianess of such graphs, are established. Secondly, the structure of and#915;(Zn) and newlineand#915;(Zn[i]), where n gt 1, is studied. We emphasize studying certain structural properties newlinesuch as the determining number and the metric dimension of and#915;(R) of those rings newlineR. In addition, the same structural properties of finite semisimple rings are also newlineexamined. newlineFurther, certain combinatorial properties in connection with the determining newlinenumber as well as the metric dimension of various classes of graphs are scrutinized newlineand consequently, the difference between the determining number and the metric newlinedimension of the zero-divisor graphs of certain commutative rings is studied. In newlinethe thesis, a 16-year-old problem by Boutin [23], regarding the possible difference newlinebetween the determining number and the metric dimension of graphs, is settled. The newlinesize of determining number of the infinite zero-divisor graph of commutative ring, is newlinetaken up for study and proved that for an infinite Noetherian ring R, the determining newlinenumber of and#915;(R) is infinite. newlineFinally, the notion of group vertex magic graphs is introduced and studied. It newlineis a new concept in graph labeling, where we label the vertices of a graph with non-zero elements of an Abelian group with respect to certain conditions. We study newlinethe possibility of labeling all trees T with diameter t gt 1, with a given Abelian group. newlineFor any finite Abelian group A , we characterize all the trees with diam tgt5.