THE ZAGREB INDICES OF BIPARTITE GRAPHS WITH MORE EDGES

Abstract

For a (molecular) graph, the first and second Zagreb indices (M₁ and M₂) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajstić. The first Zagreb index M₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let $K_{n_1,n_2}^{P}$ with n₁ $\leq$ n₂, n₁ + n₂ = n and p < n₁ be the set of bipartite graphs obtained by deleting p edges from complete bipartite graph K_n1,n2. In this paper, we determine sharp upper and lower bounds on Zagreb indices of graphs from $K_{n_1,n_2}^{P}$ and characterize the corresponding extremal graphs at which the upper and lower bounds on Zagreb indices are attained. As a corollary, we determine the extremal graph from $K_{n_1,n_2}^{P}$ with respect to Zagreb coindices. Moreover a problem has been proposed on the first and second Zagreb indices.

1. Introduction

We only consider finite, undirected and simple graphs throughout this paper. Let G be a graph with vertex set V (G) and edge set E(G). For any vertex v ∈ V (G), we denote by NG(v) the set of its neighbors in G. The degree of v ∈ V (G), denoted by dG(v), is the cardinality of NG(v), i.e., the number of vertices in G adjacent to v. For a subset W of V (G), let G-W be the subgraph of G obtained by deleting the vertices of W and the edges incident with them. Similarly, for a subset E′ of E(G), we denote by G - E′ the subgraph of G obtained by deleting the edges of E′. If W = {v} and E′ = {xy}, the subgraphs G - W and G - E′ will be written as G - v and G - xy for short, respectively. Kn1,n2 is a complete bipartite graph of order n = n1 + n2 and two bipartite sets V1 and V2 with |Vi| = ni for i = 1, 2. Other undefined notations and terminology on the graph theory can be found in [4].

A graphical invariant is a number related to a graph which is a structural invariant, in other words, it is a fixed number under graph automorphisms. In chemical graph theory, these invariants are also known as the topological indices. Two of the oldest graph invariants are the well-known Zagreb indices first introduced in [17] where Gutman and Trinajsti´c examined the dependence of total π-electron energy on molecular structure and elaborated in [18]. For a (molecular) graph G, the first Zagreb index M1(G) and the second Zagreb index M2(G) are, respectively, defined as follows:

Another well-known version of first Zagreb index is in the following:

These two classical topological indices reflect the extent of branching of the molecular carbon-atom skeleton [3,22,25]. The main properties of M1 and M2 were summarized in [5,7–9,14–16,20,24,26,28]. In particular, Deng [9] gave a unified approach to determine extremal values of Zagreb indices for trees, unicyclic, and bicyclic graphs, respectively. For some newest applications of Zagreb indices of graphs, please see [6,13,14,19,23]. In recent years, some novel variants of ordinary Zagreb indices have been introduced and studied, such as Zagreb coindices [1,2,10], multiplicative Zagreb indices [12,24,30], multiplicative sum Zagreb index [11,27] and multiplicative Zagreb coindices [29]. Especially the first and second Zagreb coindices of graph G are defined [1,10] in what follows:

Hereafter we always assume that n1, n2, p are three positive integers such that n1 ≤ n2, n1 + n2 = n and p < n1. We denote by the set of bipartite graphs obtained by deleting p edges from the complete bipartite graph Kn1,n2 . In this paper we present sharp upper and lower bounds on the Zagreb indices of graphs from and characterize the extremal graphs at which the upper or lower bounds are attained. As a corollary, we also determine the extremal graph from with respect to Zagreb coindices. Finally an open problem is proposed on the Zagreb indices.

 

2. Preliminaries

In this section we list or prove some lemmas as preliminaries, which will be further used .

Lemma 2.1 ( [1,2]). Let G be a connected graph of order n and with m edges.

Then we have

Lemma 2.2. Let G be a connected graph with e = uv ∈ E(G) and G′ = G-uv.

Then we have M1(G′) = M1(G) - 2 - 2(dG′ (u) + dG′ (v)).

Proof. By the definition of first Zagreb index, we have

which completes the proof. □

Lemma 2.3. Let G be a connected graph with uv ∈ E(G) and NG(u) ╲ {v} = {v1, v2,⋯,vα} and NG(v) ╲ {u} = {u1, u2,⋯,uβ}. Suppose that G′ = G-uv.

Then we have

Proof. From the definition of second Zagreb index, we have

Thus the proof this lemma was completed. □

 

3. Extremal graphs from w. r. t. Zagreb indices

In this section we will consider the extremal graphs from with respect to Zagreb indices. Before presenting the main results, we first introduce some special graphs in . Let be a bipartite graph obtained by deleting p edges e1, e2,⋯,ep from Kn1,n2 where all e1, e2,⋯,ep are pairwise independent. And we denote by the bipartite graph obtained by deleting p edges e1, e2,⋯,ep from Kn1,n2 where e1, e2,⋯,ep have a common vertex in the partite set of size n1 in it. Similarly, is a bipartite graph obtained by deleting p edges e1, e2,⋯,ep from Kn1,n2 where all e1, e2,⋯,ep have a common vertex in the partite set of size n2 in it. As three examples, are shown in Figure 1.

Figure 1.The graphs

When p = 1, there is only one graph in , and there is nothing to deal with for our main problem. So in what follows, we always assume that p ≥ 2. In the following theorem we will determine the extremal graphs from with respect to the first Zagreb index.

Theorem 3.1. For any graph , we have

with left equality holding if and only if and right equality holding if and only if for i = 1, 2.

Proof. We prove this result by induction on p, i.e., the number of edges deleted from Kn1,n2 . When p = 2, there exist exactly three graphs in the set , which are just . From the definition of first Zagreb index, we have

Therefore the results in this theorem hold immediately.

Assume that the results hold for p = k - 1. Now we consider the case when p = k. For any graph , there exists a graph with uv ∈ E(G∗) and G∗ - uv = G. By Lemma 2.2, we have

Now we assume that, at vertices u ∈ V1 and v ∈ V2 in G∗, there are k1, k2 edges, respectively, deleted from Kn1,n2 . Then we claim that 0 ≤ k1+k2 ≤ k-1 and dG∗ (u)+dG∗ (v) = n-k1-k2. Considering the facts that dG(u) = dG∗ (u)-1 and dG(v) = dG∗ (v) - 1, we have

Combining these two equalities (3) and (4), we arrive at the following:

Next it suffices to deal with the equality (5). For the left part in (2), by the induction hypothesis and equality (5), we have

The above equality holds if and only if and k1 = 0, k2 = 0. Equivalently, and ek = uv is independent of any one edge from {e1, e2,⋯,ek−1}. Therefore we have . Then the proof of the left part in (2) is completed.

Now we turn to the right part of (2). By the induction hypothesis and equality (5), we have

The above equality holds if and only if for i = 1, 2 and k1+k2 = k-1, i.e., and k1 = k-1, k2 = 0 or and k1 = 0, k2 = k - 1.

Thus we find that, either in , there are k - 1 edges deleted from the vertex u ∈ V1 of Kn1,n2 ; or in , there are k - 1 edges from v ∈ V2 of Kn1,n2 . Therefore, we conclude that . This completes the proof of this theorem. □

In the theorem below we characterize the extremal graphs from with respect to the second Zagreb index.

Theorem 3.2. For any graph with n1 < n2, we have

with left equality holding if and only if and right equality holding if and only if .

Proof. We prove this result by induction on p. When p = 2, there exist exactly three graphs in the set , which are just . From the definition of second Zagreb index, we have

Obviously, . Thus our results hold as desired.

Now we assume that the results in (6) hold for p = k-1. Then we consider the case when p = k. For any graph , there exists a graph with u ∈ V1, v ∈ V2, uv ∈ E(G∗) and G∗ - uv = G. The structure of G∗ is shown in Fig. 2 where the polygonal lines denote the deleted edges from Kn1,n2 . Suppose that NG∗ (u) ╲ {v} = {v1, v2,⋯,vα} ≜ X1 and NG∗ (v) ╲ {u} = {u1, u2,⋯,uβ} ≜ X4. Let V1 ╲ (X4 ∪ {u}) = X3 and V2 ╲ (X1 ∪ {v}) = X2. By Lemma 2.3, we have

Figure 2.The structure of graph G∗

As introduced in [8], for any vertex v in a graph G, we denote by mG(v) the average of the degrees of all vertices adjacent to vertex v in G. Again we assume that, at vertices u ∈ V1 and v ∈ V2 in G∗, there are k1; k2 edges, respectively, deleted from Kn1,n2 . Let the number of edges deleted between the two subsets X1,X3 in G∗ and between the two subsets X1,X4 be x1 and y1, respectively, the edges deleted between the two subsets X2,X3 and between the two subsets X2;X4 be x2 and y2, respectively. Moreover we have x1 + x2 + y1 + y2 = k - 1 - k1 - k2. Then we claim that

From the definition of mG∗ (u), we arrive at:

Similarly, we have

Combining the above two equalities with equality (7), we get

It can be easily checked that the term 2k1n1 + 2k2n2 - k1k2 reaches its minimum value 0 when k1 = k2 = 0. For the left part in (6), from equality (∗) and the induction hypothesis, we have

The above equality holds if and only if and k1 = 0, k2 = 0. Moreover, from the statement k1 = 0, k2 = 0 we can deduce that ek = uv is independent of any edge of {e1, e2,⋯,ek−1}. Therefore we find that , which ends the proof of left part in (6).

Now we will turn to the proof for the right part in (6). From the definition of mG(v) for any vertex v in a graph G and the structure of G∗, we have

Similarly, we have

Combining the above two inequalities with equality equality (7), we can obtain

Clearly the term 2(n2 -1)k2 +2(n1 -1)k1 -k1k2 reaches its maximum value 2(n2 -1)(k - 1) when k1 = 0 and k2 = k -1. From the induction hypothesis, it follows that

The above two equalities holds if and only if and k1 = 0, k2 = k - 1. That is to say, G is obtained by deleting from one more edge which has one common vertex with that one of {e1, e2,⋯,ek−1} in it. Therefore we claim that , finishing the proof of right part in (6). Thus we complete the proof of this theorem. □

Note that if n1 = n2. We denote by this graph when n1 = n2. By a similar reasoning as that in the proof of Theorem 3.2, the following corollary can be easily obtained.

Corollary 3.1. For any graph , we have

with left equality holding if and only if and right equality holding if and only if .

Now we turn to the determination of extremal graphs from with respect to Zagreb coindices. Based on Lemma 2.1 (1), we have

Moreover the following result can be easily obtained.

Corollary 3.2. For any graph , we have

with left equality holding if and only if for i = 1, 2 and right equality holding if and only if .

In view of Lemma 2.1 (2), we have

Corollary 3.3. For any graph , we have

with left equality holding if and only if and right equality holding if and only if .

Proof. From Lemma 2.1 (2), it suffices to find the extremal graphs from at which the maximal (or minimal, resp.) first and second Zagreb indices are simultaneously attained. Note that, from Theorems 3.1 and 3.2, the first Zagreb index and second Zagreb index of graphs from with n1 < n2 reach the maximum only when . Thus our results follow immediately from Theorems 3.1 and 3.2. □

 

4. A related problem

In this section we propose a problem related to the extremal graphs with respect to Zagreb indices. Based on the alternative formula (1) of the first Zagreb index and the definition of the second Zagreb index, these two indices have very similar versions. Therefore, from the intuition, we think that, in a given set G of connected graphs, the graphs with maximal first Zagreb index are the same as the graphs with maximal second Zagreb index, and vice versa; and the graphs with minimal first Zagreb index are the same as the graphs with minimal second Zagreb index, and vice versa. We say that this set G satisfies extremal identical graph property with respect to Zagreb indices (EIG property w. r. t. Zagreb indices for short). Actually, our statement is true for many known results, such as trees, unicyclic graphs, and bicyclic graphs (see [9]), and so on. Furthermore, our main result in this paper is also a positive example to our statement given above.

Now we would like to propose an interesting problem related to the EIG property as follows:

Problem 1. Characterizing the sets Γ of graphs which satisfy EIG property w. r. t. Zagreb indices?

Moreover, it is reasonable to restrict the consideration to the cases when the set Γ contains connected graphs of the same order.

Obviously, from Lemma 2.1, if a set Γ satisfies EIG property w. r. t. Zagreb indices, then it also satisfies EIG w. r. t. Zagreb coindices. Moreover we can also study the EIG property of any set of graphs with respect to other vertexdegree-based topological indices, which may be of interest to us.