On Harary index

The k-th power of a graph G, denoted by G k , is a graph with the same set of vertices as G, such that two vertices are adjacent in G k if and only if their distance in G is at most k. The Harary index H is the sum of the reciprocal distances of all pairs of vertices of the underlying graph. Lower and upper bounds on H(G k ) are obtained. A Nordhaus-Gaddum type inequality for H(G k ) is also established.

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“…(Zhou et al [13]) Let G be a connected graph of order n ≥ 5, having a connected complement G. If diam(G) = 2, then…”

Section: Theorem 12 (An Andmentioning

confidence: 99%

Harary index of the k-th power of a graph

Xiong

Gutman

2013

APPL ANAL DISCRETE M

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“…Since then, several other variants of Harary index have been introduced in the literature [22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Contribution of Modified Harary Index to Predict Kováts Retention Indices for a Set of PAHs

et al. 2016

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“…To deal with this inconsistency, the new index was proposed which was named Harary index ( [6,8]). Harary index is defined as the [5,9,11,12]). But it turned out that this modification of Wiener index has not solved the inconsistency as expected.…”

Section: Introductionmentioning

confidence: 99%

Extremal unicyclic graphs with respect to additively weighted Harary index

Sedlar¹

2015

Miskolc Math Notes

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Abstract. In this paper we define cycle-star graph CS k;n k to be a graph on n vertices consisting of the cycle of length k and n k leafs appended to the same vertex of the cycle. Also, we define cycle-path graph CP k;n k to be a graph on n vertices consisting of the cycle of length k and of path on n k vertices whose one end is linked to a vertex on a cycle. We establish that cyclestar graph CS 3;n 3 is the only maximal graph with respect to additively weighted Harary index among all unicyclic graphs on n vertices, while cycle-path graph CP 3;n 3 is the only minimal unicyclic graph (here n must be at least 5): The values of additively weighted Harary index for extremal unicyclic graphs are established, so these values are the upper and the lower bound for the value of additively weighted Harary index on the class of unicyclic graphs on n vertices.

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On Harary index

Cited by 68 publications

References 16 publications

Harary index of the k-th power of a graph

Harary index of the k-th power of a graph

Contribution of Modified Harary Index to Predict Kováts Retention Indices for a Set of PAHs

Extremal unicyclic graphs with respect to additively weighted Harary index

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