Some properties of eccentrical graphs

In this paper, the concept of the eccentrical graph of a graph is introduced. Let G be a connected graph with the vertex set V (G). The eccentrical graph of G is the graph (G) with the vertex set V ( (G) ) = V (G) and two vertices vi, vj ∈ V ( (G) ) are adjacent in (G) if and only if the distance between them is mine (vi), e (vj), where e (vi) is the eccentricity of vi. A sufficient condition for the eccentrical graph of a connected graph to be connected is given. It is proved that the eccentrical graph of every tree is connected and its diameter does not exceed 3. The extremum values of the greatest eigenvalue of eccentrical graphs of trees and connected graphs of fixed order are also studied. Furthermore, spectra of eccentrical graphs of various classes of graphs are computed.