SOME NEW RESULTS ON IRREGULARITY OF GRAPHS

Abstract

Suppose G is a simple graph. The irregularity of G, irr(G), is the summation of imb(e) over all edges $uv=e{\in}G$, where imb(e) = |deg(u)-deg(v)|. In this paper, we investigate the behavior of this graph parameter under some old and new graph operations.

1. Introduction

The degree−based graph invariants are parameters defined by degrees of vertices. The first of such graph parameters was introduced by Gutman and Trinajstić [10]. Suppose G is a graph e = uv ∈ E(G). The first Zagreb index of G is defined as M1(G) = Σv∈V (G) deg(v)2. There are a lot of works on Zagreb group invariants and interested readers can be consulted [3,8,11,16,17] for more information on this topic. The imbalance of e is defined as imb(e) = |deg(u)−deg(v)|. The summation of imbalances over all edges of G is denoted by irr(G). Albertson [2], named this parameter “irregularity” of the graph G. After this paper, there was a lot of research considering the irregularity index, see [12,14,15] for details. It is easy to see that M1(G) = Σe=uv∈E (G) [deg(u) + deg(v)]. Fath-Tabar [9], unaware from the seminal paper of Albertson and because of similarity between M1 and irr used the term “third Zagreb index” for “irregularity”.

Albertson [2] computed the maximum irregularity of various classes of graphs. As a consequence, he proved that the irregularity of an arbitrary graph with n vertices is less than and this bound is tight. Some of the present authors [20], characterized the graphs with minimum and maximum values of irregularity. Luo and Zhou [18] determined the maximum irregularity of trees and unicyclic graphs with a given number of vertices and matching number. They also characterized the extremal graphs with the mentioned property. Zhou and Luo [22], established an upper bound for irr(G) in terms of n,m, and r, where n is the order of G, m ≥ 1 is its size, and G is assumed to have no complete subgraph of order r + 1 where 2 ≤ r ≤ n − 1. They also provided new upper bounds for the irregularity of trees and unicyclic graphs. These are both functions of the number of pendant vertices of the graph under consideration. For each of these three inequalities, the authors supplied a characterization of all graphs which attain the bound. Henning and Rautenbach [15] obtained the structure of bipartite graphs having maximum possible irregularity with given cardinalities of its bipartition and given number of edges. They derived a result for bipartite graphs with given cardinalities of its bipartition and presented an upper bound on the irregularity of these graphs. In particular, they shown that if G is a bipartite graph of order n with a bipartition of equal cardinalities, then while if G is a bipartite graph with partite sets of cardinalities n1 and n2, where n1 ≥ 2n2, then irr(G) ≤ irr(Kn1,n2).

Abdo and Dimitrov [1] introduced the concept of total irregularity of a graph G and obtain some exact formula for computing total irregularity of some old graph operations. The aim of this paper is to compute formulas for the regularity of graphs under some old and new graph invariants.

Throughout this paper the path, complete and star graphs of order n are denoted by by Pn, Kn and Sn, respectively. The degree of a vertex v is denoted by degG(v). We denote by Δ(G) the maximum degree of vertices of G.

Lemma 1.1. Let G be a connected graph on n, n > 2 vertices. If G has exactly k pendant vertices then irr(G) ≽ k, with equality if and only if

Proof. Suppose u is a pendant vertex of G and v ∈ V (G) is a vertex adjacent to u. Since |V (G)| > 2, deg(v) ≽ 2 and so |deg(u)−deg(v)| ≽ 1. But, G has exactly k pendant vertices, and so irr(G) ≽ k. Notice that irr(G) = k if and only if degG(u) = 1 or 2, for each vertex u ∈ V (G). This implies that

Corollary 1.2. Let T be a tree with Δ(T) > 1. Then irr(T) ≽ Δ(T).

Proof. By assumption, T has at least Δ(T) pendant vertices and so, by Lemma 1.1, irr(T) ≽ Δ(T).

Lemma 1.3. Let T be a tree with n > 2 vertices. Then

and the lower bound is attained if and only if The upper bound is attained if and only if

Proof. Since T is a tree with n > 2 vertices, it has at least two pendant vertices and so, by lemma 1.1, 2 ≼ irr(T). On the other hand, it is not difficult to check that for each uv ∈ E(T), |deg(u) − deg(v)| ≼ n − 2 and E(T) = n − 1. So, irr(T) ≼ (n − 1)(n − 2). Notice that Sn is the unique graph with n − 2 as difference of degrees between adjacent vertices. But by Lemma 1.1, Pn is the unique tree that its third Zagreb index is equal to 2, as desired.

 

2. Main results

The join G = G1 + G2, Figure 1, of graphs G1 and G2 with disjoint vertex sets V1 and V2 and edge sets E1 and E2 is the graph union G1 ∪ G2 together with all the edges joining V1 and V2. For example, The composition G = G1[G2] of graphs G1 and G2 with disjoint vertex sets V1 and V2 and edge sets E1 and E2 is the graph with vertex set V1×V2 and u = (u1, v1) is adjacent with v = (u2, v2) whenever (u1 is adjacent to u2) or (u1 = u2 and v1 is adjacent to v2) [13]. For instance,

FIGURE 1.The Join of

The Strong product G ☒ H, Figure 2, of graphs G and H has the vertex set V (G ☒ H) = V (G) × V (H) and (a, x)(b, y) is an edge of G ☒ H if a = b and xy ∈ E(H), or ab ∈ E(G) and x = y, or ab ∈ E(G) and xy ∈ E(H). As an example, Cn ☒ K2 is the closed fence. The tensor product G ⊗ H, Figure 3, is defined as the graph with vertex set V (G) × V (H) and E(G⊗H) = {(u1, u2)(v1, v2) | u1v1 ∈ E(G) and u2v2 ∈ E(H)}. For example, Cn⊗P2 = C2n. The corona product GoH, Figure 4, is obtained by taking one copy of G and |V (G)| copies of H; and by joining each vertex of the i-th copy of H to the i-th vertex of G, 1 ≤ i ≤ |V (G)| [19]. For example, Finally, for a connected graph G, R(G) is a graph obtained from G by adding a new vertex corresponding to each edge of G, then joining each new vertex to the end vertices of the corresponding edge [21]. For example, we can see R(P2) = K3.

FIGURE 2.The Strong Product of C5 and K2.

FIGURE 3.The Tensor Product of Two Graphs.

FIGURE 4.The Corona Product of Two Graphs.

It is well-known that the Cartesian product of graphs can be recognized efficiently, in time O(mlogn) for a graph with n vertices and m edges [22]. This operation is commutative and associative as an operation on isomorphism classes of graphs, but it is not commutative. The Cartesian product is not a product in the category of graphs, but the tensor product is the categorical product.

Suppose G and H are graphs with disjoint vertex sets. Following Došslić [8], for given vertices y ∈ V (G) and z ∈ V (H) a splice of G and H by vertices y and z, (G · H)(y; z), is defined by identifying the vertices y and z in the union of G and H. Similarly, a link of G and H by vertices y and z is defined as the graph (G ∼ H)(y; z) obtained by joining y and z by an edge in the union of these graphs. Let H is a tree of progressive degree p and generation r that whose root vertex is r1. Also, let DDp,r be the graph of the regular dicentric dendrimer, Figure 6. So, it is clear that DDp,r = (H ∼ H)(r1; r1).

FIGURE 5.The Molecular Graph of Octanitrocubane.

FIGURE 6.Regular Dicentric (DD2.4) Dendrimer.

Lemma 2.1. Suppose G and H are rooted graphs with respect to the rooted vertices of a and b, respectively. Then

and the upper bound is attained if and only if for every vertex u ∈ V (G) that ua ∈ E(G), degG(a) ≽ degG(u) and for every vertex v ∈ V (H) that vb ∈ E(H), degH(b) ≽ degH(v). Moreover, the lower bound is attained if and only if for each edge au ∈ E(G), degG(u) − degG(a) ≽ degH(b) and for each edge bv ∈ E(H), degH(v) − degH(b) ≽ degG(a).

Proof. This follows immediately from the definition of splice of two graphs.

In a similar way, by definition of link of two graphs, we have:

Lemma 2.2. Suppose G and H are rooted graphs with respect to the rooted vertices of a and b, respectively. Then

and the upper bound is attained if and only if for every vertex u ∈ V (G) that ua ∈ E (G), degG(a) ≽ degG(u) and for every vertex v ∈ V (H) that vb ∈ E (H), degH(b) ≽ degH(v). Moreover, the lower bound is attained if and only if for every vertex u ∈ V (G) that ua ∈ E (G), degG(a) < degG(u) and for every vertex v ∈ V (H) that vb ∈ E (H), degH(b) < degH(v).

Lemma 2.3. Let G be a connected graph. Then irr(G) = 0 if and only if G is regular.

The line graph L(G) of a graph G is defined as follows: each vertex of L(G) represents an edge of G, and any two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint in G [21].

Lemma 2.4. Let G be a connected graph. Then irr(L(S(G))) = irr(G).

Proof. We assume that uv is an edge of L(G) that u and v are vertices corresponding to edges wx and wy of G, respectively. Then |degL(G)(u)−degL(G)(v)| = |degG(x) − degG(y)|. To compute the third Zagreb index of L(G), it is enough to calculate the summation of all degree differences of vertices of distance 2 in L(G). Therefore, irr(L(S(G))) = irr(G).

In the next result, the relationship between strong and tensor products of two graphs under the third Zagreb index is investigated.

Lemma 2.5. Let G and H be graphs. Then

Proof. The summation of |degG☒H(u) − degG☒H(v)| over all edges (a, x)(b, y) such that (a = b and xy ∈ E(H)) or (x = y and ab ∈ E(G)), is equal to:

On the other hand, for each edge (a, x)(b, y) of G ☒ H that ab ∈ E(G) and xy ∈ E(H), we have:

and hence

which completes the proof.

Lemma 2.6. Let G and H be two connected graphs. Then

Proof. Let Hi be the i-th copy of H, 1 ≤ i ≤ |V (G)|, and let G′ be the copy of G in GoH. We partition edges of GoH into the following three subsets:

If u′v′ ∈ A, then degGoH(u′) = degH(u) + 1 and degGoH(v′) = degH(v) + 1 that u′,v′ ∈ V (Hi) are corresponding to u, v ∈ V (H), respectively. Thus |degGoH(u′) − degGoH(v′)| = |degH(u) − degH(v)|. Since |V (G)| edges u′v′ in E(GoH) are corresponding to each edge uv ∈ E(H), irr1 = Σuv∈A |degGoH(u)− degGoH(v)| = |V (G)|irr(H).

It is clear that for each u′v′ ∈ B, degGoH(u′) = degG(u) + |V (H)| and degGoH(v′) = degG(v) + |V (H)|, where u′,v′ ∈ V (G′) are corresponding to u, v ∈ V (G), respectively. Hence irr2 = Σuv∈B|degGoH(u) − degGoH(v)| = irr(G).

Finally, if u′v′ ∈ C, then degGoH(u′) = degG(u) + |V (H)| and degGoH(v′) = degH(v) + 1, where u′ ∈ V (G′); v′ ∈ V (Hi) are corresponding to u ∈ V (G), v ∈ V (H), respectively. Hence |degGoH(u′) − degGoH(v′)| = degG(u) + |V (H)| − degH(v) − 1. Consequently,

By summation of irr1, irr2 and irr3, the result can be proved.

As in the proof of Lemma 2.6, |degGoH(u′)−degGoH(v′)| = degG(u)+|V (H)|− degH(v) − 1, where for each edge u′v′ ∈ E(GoH), vertices u′ ∈ V (G′) and v′ ∈ V (Hi) are corresponding to u ∈ V (G) and v ∈ V (H), respectively. Thus,

Corollary 2.7. Let G and H be two connected graphs. Then

and the upper bound is attained if and only if H is a tree and Moreover, the lower bound is attained if and only if G is a tree and

Let Gi = (Vi,Ei) be N graphs with each vertex set Vi, 1 ≤ i ≤ N, having a distinguished or root vertex, labeled 0. The hierarchical product H = GN ⊓ ... ⊓ G2 ⊓ G1 is the graph with vertices the N−tuples xN...x3x2x1, xi ∈ Vi, and edges defined by the adjacencies:

We encourage the reader to consult [5,6] for the mathematical properties of this new graph operation.

Lemma 2.8. Let G and H be connected rooted graphs and r is the root vertex of H. Then

The upper bound is attained if and only if for each ur ∈ E(H), degH(r) ≽ degH(u). Moreover, the lower bound is attained if and only if for each ur ∈ E(H) and v ∈ V (G), degH(r) + degG(v) ≼ degH(u).

Proof. Let ur ∈ E(H) and v ∈ V (G), then (v, r)(v, u) ∈ E(G ⊓ H) and so

Consequently, if degH(r) ≽ degH(u) then |degG⊓H((v, r)) − degG⊓H((v, u))| = (degH(r) − degH(u)) + degG(v) and if degH(r) + degG(v) ≼ degH(u) then |degG⊓H((v, r))−degG⊓H((v, u))| = (degH(u) −degH(r))−degG(v). On the other hand for each edge (u, r)(v, r) of G⊓H that uv ∈ E(G), we have |degG⊓H((u, r))− degG⊓H((v, r))| = |degG(u) − degG(v)| and for each edge (w, u)(w, v) of G ⊓ H that u ≠ v ≠ r, we have |degG⊓H((w, u)) − degG⊓H((w, v))| = |degH(u) − degH(v)|, which proves the result.

In what follows, let for each i, j ∈ {0, 1, 2, ...}, that i − j = 1. Let G1,G2,...,Gn be connected rooted graphs with root vertices r1,r2,...,rn, respectively. We set Also, if G = Gn ⊓ ... ⊓ G2 ⊓ G1 then we will use degG(r) to denote degG1 (r1) + degG1 (r2) + ... + degGn(rn).

Corollary 2.9. Let G1,G2, ...,Gn be connected rooted graphs with root vertices r1, r1, ..., rn, respectively. Then

The upper bound is attained if and only if for each uri ∈ E(Gi), degGi (ri) ≽ degGi (u), i = 1, 2, ..., n − 1. Moreover, the lower bound is attained if and only if for each uri ∈ E(Gi) and v ∈ V (Gi), i = 1, 2, ..., n − 1, j = i + 1, i + 2, ..., n.

Proof. Use induction on n. By Lemma 2.8, the result is valid for n = 2. Let n ≽ 3 and assume the result holds for n. Set G = Gn ⊓ ... ⊓ G2 ⊓ G1. Thus Gn+1 ⊓ ... ⊓ G2 ⊓ G1 = Gn+1 ⊓ G. Then by our assumption,

On the other hand, again by our assumption,

which completes our argument.

Example 2.10. Octanitrocubane is the most powerful chemical explosive with formula C8(NO2)8), Figure 5. Let Γ be the graph of this molecule. Then obviously Γ = Q3 ⊓ P2. If r is the root vertex of P2, one can easily see that and for each and so, by Lemma 2.8, we have

Example 2.11. Dendrimers are branched molecules have a high degree of molecular uniformity. The molecular graph of this molecules is constructed from a core and some branches connecting to the core. Let DDp,r be the graph of the regular dicentric dendrimer, see [7] for more information. Then DDp,r = P2 ⊓ H, where H is a tree of progressive degree p and generation r, Figure 6. One can see that irr(P2) = 0, irr(H) = pr+1+p, and for each ur ∈ E(H) and Therefore, by Lemma 2.8, we have:

Example 2.12. Consider the graph S4 whose root vertex is 0. If then by Lemma 2.8 we have:

and if then again by Lemma 2.8 we have:

Lemma 2.13. Let G and H be connected graphs. Then

Proof. 1). Let G be regular. Clearly, for each vertex (u, v) ∈ V (G[H]), degG[H]((u, v)) = |V (H)|degG(u) + degH(v). Since G is regular, for each edge (u, x)(v, y) ∈ G[H] that uv ∈ E(G), we have |degG[H]((u, x))−degG[H]((v, y))| = |degH(x) − degH(y)| and for every edge (u, x)(u, y) that xy ∈ E(H), we have |degG[H]((u, x))−degG[H]((u, y))| = |degH(x)−degH(y)|, which proves the result.

2). Let H be regular. Then for every edge (u, x)(u, y) that xy ∈ E(H), we have |degG[H]((u, x))−degG[H]((u, y))| = 0 and for every edge (u, x)(v, y) of G[H] that uv ∈ E(G), we have |degG[H]((u, x))−degG[H]((v, y))| = |V (H)||degG(u)− degG(v)|. On the other hand, to each edge uv ∈ E(G), there correspond |V (H)|2 edges (u, x)(v, y) in E(GoH), so irr(G[H]) = |V (H)|3irr(G).