On the adjacent eccentric distance sum of graphs. (English) Zbl 1308.05042
Summary: In this paper we show bounds for the adjacent eccentric distance sum of graphs in terms of Wiener index, maximum degree and minimum degree. We extend some earlier results of H. Hua and G. Yu [Int. Math. Forum 7, No. 25–28, 1289–1294 (2012; Zbl 1253.05064)].
The adjaceni eccentric distance sum index of the graph \(G\) is defined as
\[ \xi^{sv}(G)=\sum_{\upsilon \in V(G)} \frac {\varepsilon(\upsilon)D(\upsilon)}{\deg(\upsilon)}, \] where \(\varepsilon(\upsilon)\) is the eccentricity of the vertex \(\upsilon\), \(\deg(\upsilon)\) is the degree of the vertex \(\upsilon\) and \(D(\upsilon) = \sum_{u \in V(G)} d (u,\upsilon)\) is the sum of all distances from the vertex \(\upsilon\).
The adjaceni eccentric distance sum index of the graph \(G\) is defined as
\[ \xi^{sv}(G)=\sum_{\upsilon \in V(G)} \frac {\varepsilon(\upsilon)D(\upsilon)}{\deg(\upsilon)}, \] where \(\varepsilon(\upsilon)\) is the eccentricity of the vertex \(\upsilon\), \(\deg(\upsilon)\) is the degree of the vertex \(\upsilon\) and \(D(\upsilon) = \sum_{u \in V(G)} d (u,\upsilon)\) is the sum of all distances from the vertex \(\upsilon\).
MSC:
| 05C12 | Distance in graphs |
| 05C40 | Connectivity |
| 05C90 | Applications of graph theory |
| 05C35 | Extremal problems in graph theory |
Citations:
Zbl 1253.05064References:
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