GENERALIZATION ON PRODUCT DEGREE DISTANCE OF TENSOR PRODUCT OF GRAPHS

Abstract

In this paper, the exact formulae for the generalized product degree distance, reciprocal product degree distance and product degree distance of tensor product of a connected graph and the complete multipartite graph with partite sets of sizes m₀, m₁, ⋯ , m_r−1 are obtained.

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 tensor product of graphs has been extensively studied in relation to the areas such as graph colorings, graph recognition, decompositions of graphs, design theory, see [2,4,5,19,21].

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 [12]. 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.

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. This definition can be further generalized in the following way: and λ is a real number [13,14]. If λ = −1, then W−1(G) = H(G), where H(G) is Harary index of G. In the chemical literature also [26] as well as the general case Wλ were examined [10,15].

Dobrynin and Kochetova [6] and Gutman [11] independently proposed a vertex-degree-weighted version of Wiener index called degree distance, which is defined for a connected graph G as , where dG(u) is the degree of the vertex u in G. Similarly, the product degree distance or Gutman index of a connected graph G is defined as . The additively weighted Harary index(HA) or reciprocal degree distance(RDD) is defined in [3] as . Similarly, Su et al. [25] introduce the reciprocal product degree distance of graphs, which can be seen as a product-degree-weight version of Harary index . In [16], Hamzeh et al. recently introduced generalized degree distance of graphs. Hua and Zhang [18] have obtained lower and upper bounds for the reciprocal degree distance of graph in terms of other graph invariants. Pattabiraman et al. [22,23] have obtained the reciprocal degree distance of join, tensor product, strong product and wreath product of two connected graphs in terms of other graph invariants. The chemical applications and mathematical properties of the reciprocal degree distance are well studied in [3,20,24].

The generalized degree distance, denoted by Hλ(G), is defined as , where λ is a real number. If λ = 1, then Hλ(G) = DD(G) and if λ = −1, then Hλ(G) = RDD(G). Similarly, generalized product degree distance, denoted by , is defined as . If λ = 1, then and if λ = −1, then . Therefore the study of the above topological indices are important and we try to obtain the results related to these indices. The generalized degree distance of unicyclic and bicyclic graphs are studied by Hamzeh et al. [16,17]. Also they are given the generalized degree distance of Cartesian product, join, symmetric difference, composition and disjunction of two graphs. In this paper, the exact formulae for the generalized product degree distance, reciprocal product degree distance and product degree distance of tensor product G × Km0,m1, ...,mr−1 , where Km0,m1, ...,mr−1 is the complete multipartite graph with partite sets of sizes m0, m1, ⋯ , mr−1 are obtained.

The first Zagreb index is defined as and the second Zagreb index is defined as . In fact, one can rewrite the first Zagreb index as . The Zagreb indices were found to be successful in chemical and physico-chemical applications, especially in QSPR/QSAR studies, see [8,9].

If m0 = m1 = ⋯ = mr−1 = s in Km0,m1, ...,mr−1 (the complete multipartite graph with partite sets of sizes m0, m1, ⋯ , mr−1), then we denote it by Kr(s). For S ⊆ V (G), ⟨S⟩ denotes the subgraph of G induced by S. For two subsets S, T ⊂ V (G), not necessarily disjoint, by dG(S, T), we mean the sum of the distances in G from each vertex of S to every vertex of T, that is, .

 

2. Generalized product degree distance of tensor product of graphs

Let G be a connected graph with V (G) = {v0, v1, ⋯ , vn−1} and let Km0,m1, ...,mr−1, r ≥ 3, be the complete multiparite graph with partite sets V0, V1, ⋯ , Vr−1 with |Vi| = mi, 0 ≤ i ≤ r−1. In the graph G×Km0,m1, ...,mr−1, let Bij = vi × Vj , vi ∈ V (G) and 0 ≤ j ≤ r − 1. For our convenience, we write

Let . If vivk ∈ E(G), then the subgraph ⟨Bij ∪ Bkp⟩ of G × Km0,m1, ...,mr−1 is isomorphic to K|Vj|,|Vp| or a totally disconnected graph according to j ≠ p or j = p. It is used in the proof of the next lemma. The proof of the following lemma follows easily from the structure and properties of G × Km0,m1, ...,mr−1 .

Lemma 2.1. Let G be a connected graph on n ≥ 2 vertices and let Bij, Bkp ∈ ℬ of the graph G × Km0,m1, ...,mr−1, where r ≥ 3.

The proof of the following lemma follows easily from Lemma 2.1 and hence it is left to the reader. The lemma is used in the proof of the main theorem of this section.

Lemma 2.2. Let G be a connected graph on n ≥ 2 vertices and let Bij, Bkp ∈ ℬ of the graph G′ = G × Km0,m1, ...,mr−1, where r ≥ 3.

(i) If vivk ∈ E(G), then

(ii) If vivk ∉ E(G), then

(iii)

Lemma 2.3. Let G be a connected graph and let Bij in G′ = G×Km0,m1, ...,mr−1 . Then the degree of a vertex (vi, uj) ∈ Bij in G′ is dG′ ((vi, uj)) = dG(vi)(n0−mj), where

Lemma 2.4. Let n0 and q be the number of vertices and edges of Km0,m1, ...,mr−1. Then the sums and , where t and τ are the number of triangles and in Km0,m1, ...,mr−1 .

Theorem 2.5. Let G be a connected graph with n ≥ 2 vertices and let E2 be the set of edges of G which do not lie on any C3 of it. If n0 and q are the number of vertices and edges of Km0,m1, ...,mr−1, r ≥ 3, respectively, then , where t and τ are the number of triangles and in Km0,m1, ...,mr−1 .

Proof. Let G′ = G × Km0,m1, ...,mr−1. Clearly,

where S1 to S4 are the sums of the above terms, in order.

We shall calculate S1 to S4 of (1) separately.

First we compute S1. By Lemmas 2.2 and 2.3, we obtain:

Summing (2) over i = 0, 1, ⋯ , n − 1, we get:

Now by Lemma 2.4, we have

Next we compute S2. For this, initially we calculate

and E2 = E(G) - E1.

where M2(G) is the second Zagreb index of G. Note that each edge vivk of G is being counted twice in the sum, namely, vivk and vkvi.

Now summing (4) over j = 0, 1, ⋯ , r − 1, we get,

Now by Lemma 2.4, we have

Next we compute S3. By Lemmas 2.2 and 2.3, we obtain:

By Lemma 2.4 and the definition of generalized product degree distance, we have

Finally, we compute S4. By Lemmas 2.2 and 2.3, we obtai

By Lemma 2.4, we have

Using (1) and the sums S1,S2,S3 and S4 in (3),(5),(6) and (7), respectively, we have,

Using Theorem 2.5, we have the following corollaries.

Corollary 2.6. Let G be a connected graph with n ≥ 2 vertices. If each edge of G is on a C3, then , r ≥ 3.

For a triangle free graph, E2 = E(G) and hence

Corollary 2.7. If G is a connected triangle free graph on n ≥ 2 vertices, then , r ≥ 3.

If mi = s, 0 ≤ i ≤ r − 1, in Theorem 2.5, Corollaries 2.6 and 2.7, we have the following corollaries.

Corollary 2.8. Let G be a connected graph with n ≥ 2 vertices. Let E2 be the set of edges of G which do not lie on a triangle. Then , r ≥ 3.

Corollary 2.9. Let G be a connected graph with n ≥ 2 vertices. If each edge of G is on a C3, then , r ≥ 3.

Corollary 2.10. If G is a connected triangle free graph on n ≥ 2 vertices, then , r ≥ 3.

If we consider s = 1, in Corollaries 2.8, 2.9 and 2.10, we have the following corollaries.

Corollary 2.11. Let G be a connected graph with n ≥ 2 vertices. Let E2 be the set of edges of G which do not lie on a triangle. Then , r ≥ 3

Corollary 2.12. Let G be a connected graph on n ≥ 2 vertices. If each edge of G is on a C3, then , where r ≥ 3.

Corollary 2.13. If G is a connected triangle free graph on n ≥ 2 vertices, then , r ≥ 3.

 

3. Reciprocal product degree distance of tensor product of graphs

Using λ = −1 in Theorem 2.5, we have the reciprocal product degree distance of the graph G × Km0,m1, ...,mr−1 .

Corollary 3.1. Let G be a connected graph with n ≥ 2 vertices. Let E2 be the set of edges of G which do not lie on a triangle. Then

Using Corollary 3.1, we have the following corollaries.

Corollary 3.2. Let G be a connected graph with n ≥ 2 vertices. If each edge of G is on a C3, then , r ≥ 3.

Corollary 3.3. If G is a connected triangle free graph on n ≥ 2 vertices, then , r ≥ 3.

If mi = s, 0 ≤ i ≤ r − 1, in Corollaries 3.1, 3.2 and 3.3, we have the following corollaries:

Corollary 3.4. Let G be a connected graph with n ≥ 2 vertices. Let E2 be the set of edges of G which do not lie on a triangle. Then , r ≥ 3.

Corollary 3.5. Let G be a connected graph with n ≥ 2 vertices. If each edge of G is on a C3, then , r ≥ 3.

Corollary 3.6. If G is a connected triangle free graph on n ≥ 2 vertices, then , r ≥ 3.

If we consider s = 1 in Corollaries 3.4, 3.5, 3.6, we have the following corollaries.

Corollary 3.7. Let G be a connected graph with n ≥ 2 vertices and m edges. Let E2 be the set of edges of G which do not lie on a triangle. Then , r ≥ 3.

Corollary 3.8. Let G be a connected graph on n ≥ 2 vertices. If each edge of G is on a C3, then , where r ≥ 3.

Corollary 3.9. If G is a connected triangle free graph on n ≥ 2 vertices, then , r ≥ 3.

By direct calculations we obtain expressions for the values of the Harary indices of Kn and Cn. when n is even, and otherwise. Similarly, and RDD∗(Cn) = RDD(Cn) = 4H(Cn).

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, M2(Pn) = 4(n−2), M2(Cn) = 4n, and

Using Corollaries 3.8 and 3.9, we obtain the reciprocal product degree distance of the graphs Kn × Kr and Cn × Kr.

Example 1.

 

4. Product degree distance of tensor product of graphs

Using λ = 1 in Theorem 2.5, we have the product degree distance of the graph G × Km0,m1, ...,mr−1 .

Corollary 4.1. Let G be a connected graph with n ≥ 2 vertices and let E2 be the set of edges of G which do not lie on any C3 of it. If n0 and q are the numbers of vertices and edges of Km0,m1, ...,mr−1 , r ≥ 3, respectively, then , r ≥ 3.

Using Corollary 4.1, we have the following corollaries.

Corollary 4.2. Let G be a connected graph with n ≥ 2 vertices. If each edge of G is on a C3, then DD∗(G × Km0,m1, ...,mr−1 ) = 4q2 DD∗(G) + M1(G) (4q2 − n0q − 3t)+M2(G) (2q2 − 2n0t − 4τ), r ≥ 3.

Corollary 4.3. If G is a connected triangle free graph on n ≥ 2 vertices, then DD∗(G × Km0,m1, ...,mr−1 ) = 4q2 DD∗(G) + M1(G)(4q2 − n0q − 3t)+2M2(G)(2q2 − 2n0t − 4τ), r ≥ 3.

If mi = s, 0 ≤ i ≤ r − 1, in Corollaries 4.1,4.2 and 4.3, we have the following corollaries.

Corollary 4.4. Let G be a connected graph with n ≥ 2 vertices. Let E2 be the set of edges of G which do not lie on a triangle. Then , r ≥ 3.

Corollary 4.5. Let G be a connected graph with n ≥ 2 vertices. If each edge of G is on a C3, then DD∗(G×Kr(s)) = r2(r − 1)2s4DD∗(G)+M1(G)rs3(rs(r − 1)2 − r2 + 2r − 1)+M2(G)rs4(r − 1)2, r ≥ 3.

Corollary 4.6. If G is a connected triangle free graph on n ≥ 2 vertices, then DD∗(G×Kr(s)) = r2(r−1)2s4DD∗(G)+M1(G)rs3(rs(r−1)2 −r2 + 2r−1)+2M2(G)rs4(r − 1)2, r ≥ 3.

If we consider s = 1, in Corollaries 4.4, 4.5 and 4.6, we have the following corollaries.

Corollary 4.7. Let G be a connected graph with n ≥ 2 vertices. Let E2 be the set of edges of G which do not lie on a triangle. Then , r ≥ 3.

Corollary 4.8. Let G be a connected graph on n ≥ 2 vertices. If each edge of G is on a C3, then DD∗(G × Kr) = r2(r − 1)2DD∗(G) + M1(G)r(r − 1)3 + M2(G)r(r − 1)2, r ≥ 3.

Corollary 4.9. If G is a connected triangle free graph on n ≥ 2 vertices, then DD∗(G × Kr) = r2(r−1)2DD∗(G)+M1(G)r(r−1)3+2M2(G)r(r−1)2, r ≥ 3.

One can observe that and

Using Corollaries 4.8 and 4.9, we obtain the product degree distance of the following graphs.

Example 2.