Abstract
For any two distinct vertices u and v in a connected graph G, let lPu,v=lP be the length of u−v path P and the D–distance between u and v of G is defined as: dDu,v=minplP+∑∀y∈VPdeg y, where the minimum is taken over all u−v paths P and the sum is taken over all vertices of u−v path P. The D-index of G is defined as WDG=1/2∑∀v,u∈VGdDu,v. In this paper, we found a general formula that links the Wiener index with D-index of a regular graph G. Moreover, we obtained different formulas of many special irregular graphs.
Highlights
In the present work, the concept of D-distance between distinct vertices of G was considered, in which it depends on the length of a path P as well as the degrees of the vertices that lie on P [7]
It can be said that the graph G is rregular if every vertex in G has r degree, that is, deg v r, for all v ∈ V(G), otherwise the graph G is irregular
The Index of D-Distance for Some Special Graph n this section, we find the D-index of some special irregular graphs, such as the path graph Pn, n ≥ 2, the star graph Sn, n ≥ 4, the friendship graph Fn, of order odd n, n ≥ 7, the wheel graph Wn, n ≥ 6, and the bipartite complete graph Km,n, m ≠ n, m, n ≥ 2
Summary
We will mention some of the results that were found previously and others that will be established. Theorem 2 (see [16, 17]). If G1 and G2 are nontrivial disjoint connected graphs, (1) W(G1 × G2; x) 2W(G1; x)W(G2; x) − p(G1)W (G2; x) − p(G2)W(G1; x) + p(G1)p(G2). 2), 1), if n is an even, if n is an odd. Taking the derivative with respect to x of both sides and putting x 1, we get. W K2∗ Cn ⎪⎪⎪⎪⎪⎩ 1n n2 + 1, if n is odd
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