MULTIPLICATIVELY WEIGHTED HARARY INDICES OF GRAPH OPERATIONS

Abstract

In this paper, we present exact formulae for the multiplicatively weighted Harary indices of join, tensor product and strong product of graphs in terms of other graph invariants including the Harary index, Zagreb indices and Zagreb coindices. Finally, We apply our result to compute the multiplicatively weighted Harary indices of fan graph, wheel graph and closed fence graph.

1. Introduction

All the graphs considered in this paper are simple and connected. For vertices u, v ∈ V (G), the distance between u and v in G, denoted by dG (u, v), is the length of a shortest (u, v)-path in G and let dG(v) be the degree of a vertex v ∈ V (G). For two simple graphs G and H their tensor product, denoted by G × H, has vertex set V (G) × V (H) in which (g1, h1) and (g2, h2) are adjacent whenever g1g2 is an edge in G and h1h2 is an edge in H. Note that if G and H are connected graphs, then G × H is connected only if at least one of the graph is nonbipartite. The strong product of graphs G and H, denoted by G ⊠ H, is the graph with vertex set V (G) × V (H) = {(u, v) : u ∈ V (G), v ∈ V (H)} and (u, x)(v, y) is an edge whenever (i) u = v and xy ∈ E(H), or (ii) uv ∈ E(G) and x = y, or (iii) uv ∈ E(G) and xy ∈ E(H), see Fig.1.

Fig.1.Tensor and strong product of C3 and P3

The join G + H of graphs G and H is obtained from the disjoint union of the graphs G and H, where each vertex of G is adjacent to each vertex of H.

A topological index of a graph is a real number related to the graph; it does not depend on labeling or pictorial representation of a graph. In theoretical chemistry, molecular structure descriptors (also called topological indices) are used for modeling physicochemical, pharmacologic, toxicologic, biological and other properties of chemical compounds [11]. There exist several types of such indices, especially those based on vertex and edge distances. One of the most intensively studied topological indices is the Wiener index; for other related topological indices see [31].

Let G be a connected graph. Then Wiener index of G is defined as with the summation going over all pairs of distinct vertices of G. Similarly, the Harary index of G is defined as The Harary index of a graph G has been introduced independently by Plavsic et al. [20] and by Ivanciuc et al. [16] in 1993. Its applications and mathematical properties are well studied in [?,32,19]. Zhou et al. [33] have obtained the lower and upper bounds of the Harary index of a connected graph. Very recently, Xu et al. [28] have obtained lower and upper bounds for the Harary index of a connected graph in relation to χ(G), chromatic number of G and ω(G), clique number of G. and characterized the extremal graphs that attain the lower and upper bounds of Harary index. Various topological indices on tensor product, Cartesian product and strong product have been studied various authors, see [2,29,30,4,21,22,23,17,13].

Dobrynin and Kochetova [5] and Gutman [12] independently proposed a vertex-degree-weighted version of Wiener index called degree distance or Schultz molecular topological index, which is defined for a connected graph G as where dG (u) is the degree of the vertex u in G. Note that the degree distance is a degree-weight version of the Wiener index. Hua and Zhang [14] introduced a new graph invariant named reciprocal degree distance, which can be seen as a degree-weight version of Harary index, that is,

Similarly, the modified Schultz molecular topological index or Gutman index is defined as In Su et.al. [26] introduce the multiplicatively weighted Harary indices or reciprocal product-degree distance of graphs, which can be seen as a product -degree-weight version of Harray index

Hua and Zhang [14] have obtained lower and upper bounds for the reciprocal degree distance of graph in terms of other graph invariants including the degree distance, Harary index, the first Zagreb index, the first Zagreb coindex, pendent vertices, independence number, chromatic number and vertex and edgeconnectivity. Pattabiraman and Vijayaragavan [24,25] have obtained the exact expression for the reciprocal degree distance of join, tensor, strong and wreath product of graphs.

The first Zagreb index and second Zagerb index are defined as In fact, one can rewrite the first Zagreb index as Similarly, the first Zagreb coindex and second Zagerb coindex are defined as The Zagreb indices are found to have appilications in QSPR and QSAR studies as well, see [6]. For the survey on theory and application of Zagreb indices see [10]. Feng et al.[9] have given a sharp bounds for the Zagreb indices of graphs with a given matching number. Khalifeh et al. [18] have obtained the Zagreb indices of the Cartesian product, composition, join, disjunction and symmetric difference of graphs. Ashrafi et al. [3] determined the extremal values of Zagreb coindices over some special class of graphs. Hua and Zhang [15] have given some relations between Zagreb coindices and some other topolodical indices. Ashrafi et al. [1] have obtained the Zagreb indices of the Cartesian product, composition, join, disjunction and symmetric difference of graphs.

A path, cycle and complete graph on n vertices are denoted by Pn, Cn and Kn, respectively. We call C3 a triangle. In this paper, we present exact formulae for the multiplicatively weighted Harary indices of join, tensor product and strong product of graphs in terms of other graph invariants including the Harary index, Zagreb indices and Zagreb coindices. Finally, We apply our result to compute the multiplicatively weighted Harary indices of fan graph, wheel graph and closed fence graph.

 

2. Multiplicatively weighted Harary index of G1 + G2

In this section, we compute the Multiplicatively Weighted Harary Index of join of two graphs.

Theorem 2.1. Let G1 and G2 be graphs with n and m vertices p and q edges, respectively. Then

Proof. Set V (G1) = {u1, u2, . . . , un} and V (G2) = {v1, v2, . . . , vm}. By definition of the join of two graphs, one can see that,

Therefore,

Using Theorem 2.1, we have the following corollaries.

Corollary 2.2. Let G be graph on n vertices and p edges. Then

Let Kn,m be the bipartite graph with two partitions having n and m vertices. Note that

Corollary 2.3.

One can observe that M1 (Cn) = 4n, n ≥ 3, M1 (P1) = 0, M1 (Pn) = 4n − 6, n > 1 and M1 (Kn ) = n(n − 1) 2. Similarly, By direct calculations we obtain the second Zagreb indices and coindices of Pn and Cn.

Using Corollary 2.2, we compute the formulae for reciprocal degree distance of star, fan and wheel graphs, see Fig.2.

Fig. 2Fan graph and wheel graph

Example 1.

 

3. Multiplicatively weighted Harary index of tensor product of graphs

In this section, we compute the Multiplicatively weighted Harary index of G × Kr.

The proof of the following lemma follows easily from the properties and structure of G × Kr. The lemma is used in the proof of the main theorem of this section.

Lemma 3.1. Let G be a connected graph on n ≥ 2 vertices. For any pair of vertices xij, xkp ∈ V (G × Kr), r ≥ 3, i, k ∈ {1, 2, . . . , n} j, p ∈ {1, 2, . . . , r}. Then

(i) If uiuk ∈ E(G), then

(ii) If uiuk ≠ E(G), then dG×Kr (xij, xkp) = dG (ui, uk).

(iii) dG×Kr (xij, xip) = 2.

Proof. Let V (G) = {u1, u2 , . . . , un} and V (Kr) = {v1, v2 , . . . , vr} . Let xij denote the vertex (ui, vj) of G × Kr. We only prove the case when uiuk ∉ E(G), i ≠ k and j = p. The proofs for other cases are similar.

We may assume j = 1. Let P = uius1us2 . . . uspuk be the shortest path of ength p + 1 between ui and uk in G. From P we have a (xi1, xk1)-path P1 = xi1xs12 . . . xsp-12xsp3xk1 if the length of P is odd, and P1 = xi1xs12 . . . xsp-12xsp2xk1 if the length of P is even.

Obviously, the length of P1 is p + 1, and thus dG×Kr (xi1, xk1) ≤ p + 1 ≤ dG(ui, uk). If there were a (xi1, xk1)-path in G × Kr that is shorter than p + 1 then it is easy to find a (ui, uk)-path in G that is also shorter than p + 1 in contrast to dG(ui, uk) = p + 1. □

Theorem 3.2. Let G be a connected graph with n ≥ 2 vertices and m edges. Then

Proof. Set V (G) = {u1, u2, . . . , un} and V (Kr) = {v1, v2, . . . , vr}. Let xij denote the vertex (ui, vj) of G × Kr. The degree of the vertex xij in G × Kr is dG(ui) dKr (vj), that is dG×Kr (xij) = (r − 1)dG(ui). By the definition of multiplicatively weighted Harary index

where A1 to A3 are the sums of the above terms, in order.

We shall calculate A1 to A3 of (1) separately.

(A1) First we compute

(A2) Next we compute

Let E1 = {uv ∈ E(G) | uv is on a C3 in G} and E2 = E(G) − E1.

Now summing (3) over j = 0, 1, . . . , r − 1, we get,

(A3) Next we compute

Using (1) and the sums A1,A2 and A3 in (2),(4) and (5), respectively, we have,

Using Theorem 3.2, we have the following corollaries.

Corollary 3.3. Let G be a connected graph on n ≥ 2 vertices with m edges. If each edge of G is on a C3, then

For a triangle free graph

Corollary 3.4. If G is a connected triangle free graph on n ≥ 2 vertices and m edges, then

By direct calculations we obtain expressions for the values of the Harary indices of Kn and Cn. when n is even, and otherwise. Similarly,

Using Corollaries 3.3 and 3.4, we obtain the multiplicatively weighted Harary indices of the graphs Kn × Kr and Cn × Kr.

Example 2. (i)

(ii)

 

4. Multiplicatively weighted Harary index of strong product of graphs

In this section, we obtain the multiplicatively weighted Harary index of G ⊠ Kr.

Theorem 4.1. Let G be a connected graph with n vertices and m edges. Then

Proof. Set V (G) = {u1, u2, . . . , un} and V (Kr) = {v1, v2, . . . , vr}. Let xij denote the vertex (ui, vj ) of G ⊠ Kr. The degree of the vertex xij in G ⊠ Kr is dG(ui) + dKr (vj) + dG(ui )dKr (vj), that is dG⊠Kr (xij) = rdG (ui) + (r − 1). One can see that for any pair of vertices xij, xkp ∈ V (G ⊠ Kr), dG⊠Kr (xij, xip) = 1 and dG⊠Kr (xij, xkp) = dG(ui, uk).

where A1, A2 and A3 are the sums of the terms of the above expression, in order. We shall obtain A1 to A3 of (6), separately.

Using (7), (8) and (9) in (6), we have

Using Theorem 4.1, we obtain the following corollary.

Corollary 4.2.

As an application we present formula for multiplicatively weighted Harary index of closed fence graph, Cn ⊠ K2, see Fig. 3.

Fig. 3.Closed fence graph

Example 3. By Corolarry 4.2, we have