Wiener index of sum of shadowgraphs

Abstract

The Wiener index of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text] where the sum is taken through all unordered pairs of vertices of [Formula: see text] and [Formula: see text] is distance between two vertices [Formula: see text] and [Formula: see text] of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two graphs. For a graph [Formula: see text], let [Formula: see text] be a copy of [Formula: see text] and [Formula: see text]. The [Formula: see text]-sum [Formula: see text] is a graph with the set of vertices [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text], where [Formula: see text] be one of the shadowgraph [Formula: see text] or closed shadowgraph [Formula: see text]. In this paper, we reported the Wiener index of these graphs.

Interval Valued Intuitionistic Fuzzy Line Graphs

“…We start the section with basic definitions related to IVI-FLG. So that, the definitions [1][2][3][4][5][6][7][8][9][10][11][12] are the well-known definitions used to discuss the main result of this work.…”

“…After Euler was presented with the impression of Königsberg bridge problem, Graph Theory has become recognized in different academic fields like engineering, social science in medical science, and natural science. A few operations of graphs like line graph, wiener index of graph, cluster and corona operations of graph, total graph, semi-total line and edge join of graphs have been valuable in graph theory and chemical graph theory to consider the properties of boiling point, heat of evaporation, surface tension, vapor pressure, total electron energy of polymers , partition coefficients, ultrasonic sound velocity and internal energy [1][2][3][4]. The degree sequence of a graph and algebraic structure of different graphs operations were determined and its result is to the join and corona products of any number of graphs [5].…”

Application of maple on computing strong fuzzy chromatic polynomial of fuzzy graphs

“…A few operations of graphs like fuzzy graph, wiener index of graph, cluster and corona operations of graph, total graph, semi-total line and edge join of graphs have been valuable in graph theory and chemical graph theory to consider the properties of boiling point, heat of evaporation, surface tension, vapor pressure, total electron energy of polymers, partition coefficients, ultrasonic sound velocity and internal energy. Monte Carlo simulation technique is used in survival signature to analyze the fuzzy reliability of systems having complexity [26][27][28][29][30][31].…”

Application of Maple on Computing Strong Fuzzy Chromatic Polynomial of Fuzzy Graphs

“…In recent times, the notion of fuzzy chromatic polynomials and its properties [21][22] and the notion of strong fuzzy chromatic polynomial of fuzzy graphs [23] A few operations of graphs like fuzzy graph, wiener index of graph, cluster and corona operations of graph, total graph, semi-total line and edge join of graphs have been valuable in graph theory and chemical graph theory to consider the properties of boiling point, heat of evaporation, surface tension, vapor pressure, total electron energy of polymers, partition coefficients, ultrasonic sound velocity and internal energy. Monte Carlo simulation technique is used in survival signature to analyze the fuzzy reliability of systems having complexity [26][27][28][29][30][31]. The degree sequence of a graph and algebraic structure of different graphs operations were determined and its result is to the join and corona products of any number of graphs For the notations not declared in this manuscript, to understand well we recommend the readers to refer [32,33,34].…”