On the Difference Between the Eccentric Connectivity Index and Eccentric Distance Sum of Graphs
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On the Difference Between the Eccentric Connectivity Index and Eccentric Distance Sum of Graphs Yaser Alizadeh1 · Sandi Klavžar2,3,4 Received: 6 May 2020 / Revised: 25 August 2020 / Accepted: 1 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract The eccentric connectivity index of a graph G is ξ c (G) = v∈V (G) ε(v) deg(v), and the eccentric distance sum is ξ d (G) = ε(v)D(v), where ε(v) is the v∈V (G) eccentricity of v, and D(v) the sum of distances between v and the other vertices. A lower and an upper bound on ξ d (G)−ξ c (G) is given for an arbitrary graph G. Regular graphs with diameter at most 2 and joins of cocktail-party graphs with complete graphs form the graphs that attain the two equalities, respectively. Sharp lower and upper bounds on ξ d (T ) − ξ c (T ) are given for arbitrary trees. Sharp lower and upper bounds on ξ d (G) + ξ c (G) for arbitrary graphs G are also given, and a sharp lower bound on ξ d (G) for graphs G with a given radius is proved. Keywords Eccentricity · Eccentric connectivity index · Eccentric distance sum · Tree Mathematics Subject Classification 05C12 · 05C09 · 05C92
1 Introduction In this paper, we consider simple and connected graphs. If G = (V (G), E(G)) is a graph and u, v ∈ V (G), then the distance dG (u, v) between u and v is the number
Communicated by Rosihan M. Ali.
B
Sandi Klavžar [email protected] Yaser Alizadeh [email protected]
1
Department of Mathematics, Hakim Sabzevari University, Sabzevar, Iran
2
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
3
Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia
4
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
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Y. Alizadeh, S. Klavžar
of edges on a shortest u, v-path. The eccentricity of a vertex and its total distance are distance properties of central interest in (chemical) graph theory; they are defined as follows. The eccentricity εG (v) of a vertex v is the distance between v and a farthest vertex from v, and the total distance DG (v) of v is the sum of distances between v and the other vertices of G. Even more fundamental property of a vertex in (chemical) graph theory is its degree (or valence in chemistry), denoted by degG (v). (We may skip the index G in the above notations when G is clear.) Multiplicatively combining two out of these three basic invariants naturally leads to the eccentric connectivity index ξ c (G), the eccentric distance sum ξ d (G), and the degree distance D D(G), defined as follows: ξ c (G) =
ε(v) deg(v) .
v∈V (G)
ξ d (G) =
ε(v)D(v) .
v∈V (G)
D D(G) =
deg(v)D(v) .
v∈V (G)
ξ c was introduced by Sharma, Goswami, and Madan [18], ξ d by Gupta, Singh, and Madan [7], and D D by Dobrynin and Kochetova [6] and by Gutman [8]. These three topological indices are well investigated, selected contributions to the eccentric connectivity index are [10,13,24], see also [21] for its generalization; to the eccentric d
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