Additively weighted Harary index of some composite graphs

Alizadeh, Yaser; Iranmanesh, Ali; Došlić, Tomislav

doi:10.1016/j.disc.2012.09.011

Cited by 55 publications

(27 citation statements)

References 18 publications

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“…which is just the reciprocal sum-degree distance and reciprocal product-degree distance, respectively. For the research development on H A (G) and H M (G), one may be referred to [2], [9], [20], [25], [36]. The notion of the n-vertex k-tree T k n for k 1 was first introduced by Beineke and Pippert [6] in 1969 (its definition is given in the following).…”

Section: H(g)mentioning

confidence: 99%

See 1 more Smart Citation

Extremal properties of distance-based graph invariants for $k$-trees

Zhang¹,

Li²

2017

Math.Boh.

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Sharp bounds on some distance-based graph invariants of n-vertex k-trees are established in a unified approach, which may be viewed as the weighted Wiener index or weighted Harary index. The main techniques used in this paper are graph transformations and mathematical induction. Our results demonstrate that among k-trees with n vertices the extremal graphs with the maximal and the second maximal reciprocal sum-degree distance are coincident with graphs having the maximal and the second maximal reciprocal product-degree distance (and similarly, the extremal graphs with the minimal and the second minimal degree distance are coincident with graphs having the minimal and the second minimal eccentricity distance sum).

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Section: H(g)mentioning

confidence: 99%

“…For the n-vertex k-tree T k n , if k = 1, then it is just a tree and the extremal properties of W (G), H(G), DD(G), ξ d (G), H A (G), and H M (G) of n-vertex tree can be referred to in [2], [9], [10], [12], [26], [40], [41]. On the other hand, T k n ∼ = K n for k n k + 1 and T k n ∼ = P k n ∼ = S k,n−k for n = k + 2.…”

Section: Definition 12 ([12]mentioning

confidence: 99%

Extremal properties of distance-based graph invariants for $k$-trees

Zhang¹,

Li²

2017

Math.Boh.

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“…In order to close the gap between the two research communities by drawing their attention to a generalization of a concept, which gives more weight to the contributions of pairs of vertices of high degrees, recently, Alizadeh et al [1] introduced an invariant, named additively weighted Harary index, which is defined as…”

Section: Introductionmentioning

confidence: 99%

“…It is known that the intuitive idea of pairs of close atoms contributing more than the distant ones has been difficult to capture in topological indices. A possibly useful approach could be to replace the additive weighting of pairs by the multiplicative one, thus giving rise to a new invariant, named multiplicatively weighted Harary index [1]:…”

Section: Introductionmentioning

confidence: 99%

Multiplicatively weighted Harary index of some composite graphs

Xiong

2015

Filomat

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Recently, Alizadeh et al. [Discrete Math., 313 (2013): 26-34] proposed a modification of the Harary index in which the contributions of vertex pairs are weighted by the product of their degrees. It is named multiplicatively weighted Harary index and defined as: H M (G) = u v δ G (u)•δ G (v) d G (u,v) , where δ G (u) denotes the degree of the vertex u in the graph G and d G (u, v) denotes the distance between two vertices u and v in the graph G. In this paper, after establishing basic mathematical properties of this new index, we proceed by finding the extremal graphs and presenting explicit formulae for computing the multiplicatively weighted Harary index of the most important graph operations such as the join, composition, disjunction and symmetric difference of graphs.

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“…In [2] the authors increased the attenuation of contributions of vertex pairs with their distance. In [1] an attempt was made in a different direction, the authors introduced a correction that gives more weight to the contributions of pairs of vertices of high degrees. This modification was named additively weighted Harary index.…”

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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Additively weighted Harary index of some composite graphs

Cited by 55 publications

References 18 publications

Extremal properties of distance-based graph invariants for $k$-trees

Extremal properties of distance-based graph invariants for $k$-trees

Multiplicatively weighted Harary index of some composite graphs

Extremal unicyclic graphs with respect to additively weighted Harary index

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