Degree distance and Gutman index of two graph products

Distance-based indices of complete m-ary trees

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383092050041x ◽

2020 ◽

Vol 12 (04) ◽

pp. 2050041

Author(s):

Mesfin Masre ◽

Samuel Asefa Fufa ◽

Tomáš Vetrík

Keyword(s):

Computer Science ◽

Recurrence Relations ◽

Wiener Index ◽

The Other ◽

Binary Trees ◽

New Methods ◽

Degree Distance ◽

Gutman Index ◽

Eccentric Distance

Binary and [Formula: see text]-ary trees have extensive applications, particularly in computer science and chemistry. We present exact values of all important distance-based indices for complete [Formula: see text]-ary trees. We solve recurrence relations to obtain the value of the most well-known index called the Wiener index. New methods are used to express the other indices (the degree distance, the eccentric distance sum, the Gutman index, the edge-Wiener index, the hyper-Wiener index and the edge-hyper-Wiener index) as well. Values of distance-based indices for complete binary trees are corollaries of the main results.

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Computing Exact Values for Gutman Indices of Sum Graphs under Cartesian Product

Mathematical Problems in Engineering ◽

10.1155/2021/5569997 ◽

2021 ◽

Vol 2021 ◽

pp. 1-20

Author(s):

Abdulaziz Mohammed Alanazi ◽

Faiz Farid ◽

Muhammad Javaid ◽

Augustine Munagi

Keyword(s):

Connected Graph ◽

Topological Index ◽

Cartesian Product ◽

Chemical Compounds ◽

Factor Graphs ◽

Cartesian Products ◽

Degree Distance ◽

Gutman Index ◽

Extremal Theory ◽

Theory Of Graphs

Gutman index of a connected graph is a degree-distance-based topological index. In extremal theory of graphs, there is great interest in computing such indices because of their importance in correlating the properties of several chemical compounds. In this paper, we compute the exact formulae of the Gutman indices for the four sum graphs (S-sum, R-sum, Q-sum, and T-sum) in the terms of various indices of their factor graphs, where sum graphs are obtained under the subdivision operations and Cartesian products of graphs. We also provide specific examples of our results and draw a comparison with previously known bounds for the four sum graphs.

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WIENER INDEX OF TREES OF GIVEN ORDER AND DIAMETER AT MOST

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972713000816 ◽

2013 ◽

Vol 89 (3) ◽

pp. 379-396 ◽

Cited By ~ 11

Author(s):

SIMON MUKWEMBI ◽

TOMÁŠ VETRÍK

Keyword(s):

Upper Bound ◽

Open Problem ◽

Wiener Index ◽

Upper Bounds ◽

Degree Distance ◽

Gutman Index

AbstractThe long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order$n$and diameter at most$6$.

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Steiner Degree Distance of Two Graph Products

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2019-0020 ◽

2019 ◽

Vol 27 (2) ◽

pp. 83-99 ◽

Cited By ~ 1

Author(s):

Yaping Mao ◽

Zhao Wang ◽

Kinkar Ch. Das

Keyword(s):

Connected Graph ◽

Cartesian Product ◽

Graph Products ◽

Product Graphs ◽

Degree Distance ◽

Cartesian Product Graphs

AbstractThe degree distance DD(G) of a connected graph G was invented by Dobrynin and Kochetova in 1994. Recently, one of the present authors introduced the concept of k-center Steiner degree distance defined as SDD_k (G) = \sum\limits_{\mathop {S \subseteq V(G)}\limits_{\left| S \right| = k} } {\left[ {\sum\limits_{v \in S} {{\it deg} _G (v)} } \right]d_G (S),} where dG(S) is the Steiner k-distance of S and degG(v) is the degree of the vertex v in G. In this paper, we investigate the Steiner degree distance of complete and Cartesian product graphs.

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A note on Steiner reciprocal degree distance

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500500 ◽

2020 ◽

Vol 12 (04) ◽

pp. 2050050

Author(s):

D. Sarala ◽

S. K. Ayyaswamy ◽

S. Balachandran ◽

K. Kannan

Keyword(s):

Connected Graph ◽

Natural Generalization ◽

Bipartite Graphs ◽

Graph Distance ◽

Straightforward Method ◽

Steiner Formula ◽

Degree Distance ◽

Complete Bipartite Graphs ◽

Gutman Index ◽

Distance Formula

The concept of reciprocal degree distance [Formula: see text] of a connected graph [Formula: see text] was introduced in 2012. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. The [Formula: see text]-center Steiner reciprocal degree distance defined as [Formula: see text], where [Formula: see text] is the Steiner [Formula: see text]-distance of [Formula: see text] and [Formula: see text] is the degree of the vertex [Formula: see text] in [Formula: see text]. Motivated from Zhang’s paper [X. Zhang, Reciprocal Steiner degree distance, Utilitas Math., accepted for publication], we find the expression for [Formula: see text] of complete bipartite graphs. Also, we give a straightforward method to compute Steiner Gutman index and Steiner degree distance of path.

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