On the general degree-eccentricity index of a graph
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On the general degree-eccentricity index of a graph Mesfin Masre1 · Tomáš Vetrík2 Received: 31 March 2020 / Accepted: 18 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract We define the general degree-eccentricity index of a connected graph G as D E Ia,b (G) = a b v∈V (G) dG (v)eccG (v) for a, b ∈ R, where V (G) is the vertex set of G, dG (v) is the degree of a vertex v and eccG (v) is the eccentricity of v in G. Let I1 be any index which decreases with the addition of edges and let I2 be any index which increases with the addition of edges. We obtain sharp lower bounds on the I1 index and the D E Ia,b index, where a < 0 and b > 0, and sharp upper bounds on the I2 index and the D E Ia,b index, where a > 0 and b < 0, for connected graphs of given order in combination with given independence number, vertex cover number or minimum degree. We also present sharp upper bounds on the D E Ia,b index, where a ≥ 1 and b < 0, for connected graphs of given order n in combination with given vertex connectivity, edge connectivity, number of pendant vertices, number of bridges or matching number β ≤ n4 . Keywords General degree-eccentricity index · Eccentricity · Eccentric connectivity index Mathematics Subject Classification 05C12 · 05C07 · 05C35
1 Introduction Topological indices have been used and have shown to give a high degree of predictability of pharmaceutical properties. Experiments reveal that prediction using the eccentric connectivity index of analgesic activity [13] and of anti-inflammatory activity [3] is very accurate. Eccentricity-based indices provide very good correlations with respect to both, physical and biological properties of chemical substances [2]. Let G be a simple connected graph with vertex set V (G) and edge set E(G). The order of a graph G is the number of vertices of G. The degree dG (v) of a vertex v ∈ V (G) is
The work of T. Vetrík is based on the research supported by the National Research Foundation of South Africa (Grant Number 126894).
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Tomáš Vetrík [email protected]
1
Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia
2
Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa
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M. Masre , T. Vetrík
the number of edges incident to v. The distance dG (u, v) between two vertices u, v ∈ V (G) is the number of edges in a shortest path connecting them. The eccentricity of v, eccG (v), is the distance between v and any vertex furthest from v in G. The diameter of G is the distance between any two furthest vertices. Let us introduce the general degree-eccentricity index of a connected graph G as
D E Ia,b (G) =
b dGa (v)eccG (v)
v∈V (G)
for a, b ∈ R. If b = 1, we obtain the general eccentric connectivity index; see [15]. If a = 1 and b = 1, we get the basic eccentric connectivity index. If a = 1 and b = −1, we obtain the connective eccentricity index. Mathematical properties of eccentricity-based indices have been studied due to ex
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