Eccentric graph of a graph and its structural properties

Eccentricity plays a crucial role in graph theory as it quantifies the distance of a vertex from other vertices within a graph. This concept not only helps determine the diameter and radius of a graph but also enables the computation of various other graph properties. Its significance extends beyond the realm of graph theory, with extensive applications in diverse fields such as molecular graph theory, chemical networks, communication theo ry, and boiling phenomena. This thesis delves into exploring the properties of eccentric graphs, with a particular focus on those associated with extremal graphs and trees. The second chapter presents a compilation of fundamental definitions and graph-related concepts. Moving on to the third chapter, we review existing literature on eccentric graphs, examining cases for some extreme graphs and investigating unique eccentric point graphs and diameter max imal graphs. Furthermore, we address a deficiency in Akiyama's theory from 1985, where we provide necessary and sufficient conditions for the eccentric graph to be equal to the complement of the original graph. Building upon this, we develop new theories and present results concerning eccentric graphs of trees. By shedding light on the intricacies of eccentricity-based graph measurements and ex ploring novel applications, this study contributes valuable insights to the field of graph theory and its various interdisciplinary connections.