Eccentric Coloring in graphs

he \emph{eccentricity} $e(u)$ of a vertex $u$ is the maximum distance of $u$ to any other vertex of $G$. A vertex $v$ is an \emph{eccentric vertex} of vertex $u$ if the distance from $u$ to $v$ is equal to $e(u)$. An \emph{eccentric coloring} of a graph $G = (V, E)$ is a function \emph{color}: $V \rightarrow N$ such that\\ (i) for all $u, v \in V$, $(color(u) = color(v)) \Rightarrow d(u, v) > color(u)$.\\ (ii) for all $v \in V$, $color(v) \leq e(v)$.\\ The \emph{eccentric chromatic number} $\chi_{e}\in N$ for a graph $G$ is the lowest number of colors for which it is possible to eccentrically color \ $G$ \ by colors: $V \rightarrow \{1, 2, \ldots , \chi_{e} \}$. In this paper, we have considered eccentric colorability of a graph in relation to other properties. Also, we have considered the eccentric colorability of lexicographic product of some special class of graphs

A Characterization For 2-Self-Centered Graphs

A Graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their complements. Then we split characterizing edge-minimal 2-self-centered graphs into two cases. First, we characterize edge-minimal 2-self-centered graphs without triangles by introducing \emph{specialized bi-independent covering (SBIC)} and a structure named \emph{generalized complete bipartite graph (GCBG)}. Then, we complete characterization by characterizing edge-minimal 2-self-centered graphs with some triangles. Hence, the main characterization is done since a graph is 2-self-centered if and only if it is a spanning subgraph of some edge-maximal 2-self-centered graphs and, at the same time, it is a spanning supergraph of some edge-minimal 2-self-centered graphs