The \emph{eccentricity} of a vertex $u$ in a graph $G$, denoted by $e_G(u)$, is the maximum distance from $u$ to other vertices in $G$. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity indices, denoted by $\sigma_0(G)$, $\sigma_1(G)$, and $\sigma_2(G)$, respectively. These are defined by $\sigma_0(G)=\frac{1}{|V(G)|}\sum_{u\in V(G)}e_G(u)$, $\sigma_1(G)=\sum_{u\in V(G)}e_G^2(u)$, and $\sigma_2(G)=\sum_{uv\in E(G)}e_G(u)e_G(v)$. We study lower and upper bounds on these parameters among $n$-vertex connected graphs with fixed diameter, chromatic number, clique number, or matching number. Most of the bounds are sharp, with the corresponding extremal graphs characterized.

Distance in graphs

Bounds for eccentricity-based parameters of graphs

The eccentricity $ e(u) $ of a vertex $ u $ is the maximum distance from $u$ in $G$. A vertex $v$ is an eccentric vertex of $ u $ if the distance from $u$ to $v$ is equal to $e(u) $. An eccentric coloring of a graph $G = (V,E)$ is a function color: $V \longrightarrow N$ such that (i)  for all $u, v \in V$, $ (color(u) = color(v))\Longrightarrow d(u,v) > color(u) $, (ii)  for all $v \in V$, color $ (v)\leq e(v)$. The eccentric chromatic number $ \chi_{e} \in N $ for a graph $ G $ is the least number of colors for which it is possible to eccentrically color $ G $ by colors: $V \longrightarrow \lbrace1,2,....,\chi_{e}\rbrace. $ In this paper, we have proved that a cycle with a chord between vertices at any distance up to the radius of the cycle is eccentric colorable thereby making the results of [5] particular cases. Also, here we have extended results on eccentric coloring of Lexicographic product graphs proved earlier and found a sharp upper bound and shown its attainability.

background

More results on eccentric coloring in graphs

For any vertex x in a connected graph G of order p ≥ 2, a set S ⊆ V (G) is an x-monophonic set of G if each vertex v ∈ V (G) lies on an x − y monophonic path for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by mx(G). An x-monophonic set S is called a minimal x-monophonic set if no proper subset of S is an x-monophonic set. The upper x-monophonic number, denoted by m + (G), is defined as the maximum cardinality of a minimal x-monophonic set of G. We determine bounds for it and find the same for some special classes of graphs. For any two positive integers a and b with 1 ≤ a ≤ b, there exists a connected graph G with mx(G) = a and m + (G) = b for some vertex x in G. Also, it is shown that for any three positive integers a, b and n with a ≥ 2 and a ≤ n ≤ b, there exists a connected graph G with mx(G) = a, m + (G) = b and a minimal x-monophonic set of cardinality n.

/pdf/the-upper-vertex-monophonic-number-of-a-graph-3w8u5z23du.pdf

The upper vertex monophonic number of a graph

In this paper we introduce the notion of Maxideal and Minideal zero divisor graphs of the maximal and minimal ideals of the ring Zn. We defined the Maxideal and Minideal zero divisor graphs of the ring Zn for any n, to be the zero divisor graph of maximal and minimal ideals of the ring Zn. Finally we define the zero divisor ideal I*is an ideal contains of all adjacent vertices of the ideal I, and zero divisor ideal graphs of I*is denoted by ГI(Zn), is a graph with vertex set I*.

http://www.ijoar.org/journals/IJOARM/papers/THE-ZERO-DIVISOR-IDEAL-GRAPH-OF-THE-RING-ZN.pdf

The zero divisor ideal graph of the ring zn

A subset D of the vertex set V (G) of a graph G is said to be a dominating set if every vertex not in D is adjacent to at least one vertex in D. A dominating set D is said to be an eccentric dominating set if for every v ∈ V − D, there exists at least one eccentric point of v in D. An eccentric dominating set D of G is a nonsplit eccentric dominating set if the induced sub graph is connected. The minimum of the cardinalities of the nonsplit eccentric dominating sets of G is called the nonsplit eccentric domination number γnsed(G). In this paper, we have studied the changing and unchanging of Nonsplit eccentric domination number in graphs.

https://www.jetir.org/papers/JETIRR006007.pdf

Distance in graphs

Citations

Bounds for eccentricity-based parameters of graphs

More results on eccentric coloring in graphs

The upper vertex monophonic number of a graph

The zero divisor ideal graph of the ring zn

Changing and Unchanging of Nonsplit Eccentric Domination number in graphs.

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