enumeration of the number of spanning trees in some ... - Toubkal

118 CHAPTER 7. MAXIMAL PLANAR MAPSFigure 7.5: The graphs G and H are not the same, however, they are isomorphic.7.2 Calculating the Wiener index in the maximal planarmapsIn this section, we shall focus on the Wiener index in case of planar maps and give aformula which calculates the Wiener index in the maximal planar map, then we give aninequality, which minimizes and maximizes any planar map by the maximal planar mapwith n vertices and the path of n vertices; like the inequality in the trees.7.2.1 IntrodutionIn a communication network, large diameter may be acceptable if most pairs can communicatevia short paths. This leads us to study the average distance instead of themaximum. Since the average is the sum divided by ( n2)(the number of vertex pairs), itis equivalent to study D(G) = ∑ {v i ,v j }⊆V (G) d(v i, v j ). The sum D(G) has been called theWiener index of G (also written W (G)). It is used by Wiener to study the boiling pointof paraffin. Molecules can be modeled by graphs with vertices for atoms and edges foratomic bonds. Many chemical properties of molecules are related to the Wiener indexof the corresponding graphs. We study the extreme values of W (G). In the next, wefocus on the Wiener index in planar maps, the particular case of trees has been echoedby several people [39, 69, 125]. We study the extreme values of W (G). Refer to [2] and[117] for the significance of the Wiener index.7.2.2 Calculation of the Wiener index in the planar mapsHerein, we give some basic definitions and properties about the Wiener index. An importantconcept that we need in this section is that of the Wiener index in the case of planarmaps.Definition 7.2.1 The distance between two distinct vertices v i and v j of a map C, denotedby d(v i , v j ) is equal to the length of (number of edges in) the shortest path thatconnects v i and v j . Conventionally, d(v i , v i ) = 0.