Graphs with maximal σ irregularity

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Graphs with maximal σ irregularity Hosam Abdo a , Darko Dimitrov b, *, Ivan Gutman c a b c

Institut für Informatik, Freie Universität Berlin, Germany Faculty of Information Studies, Novo mesto, Slovenia Faculty of Science, University of Kragujevac, Serbia

article

a b s t r a c t

info

Article history: Received 30 November 2017 Received in revised form 20 April 2018 Accepted 6 May 2018 Available online xxxx

A natural extension of the well-known Albertson irregularity index is the recently introduced σ irregularity index.∑ For a simple graph G with an edge set E(G), the σ irregularity 2 index is defined as σ (G) = uv∈E(G) (dG (u) − dG (v )) , where dG (v ) denotes the degree of the vertex v of G. Here, we characterize general graphs with maximal σ irregularity. We also present lower bounds on the maximal σ irregularity of graphs with fixed minimal and/or maximal vertex degrees. © 2018 Elsevier B.V. All rights reserved.

Keywords: Irregularity of a graph Albertson irregularity index σ irregularity index Graphs with extremal irregularity

1. Introduction Let G be a simple undirected connected graph with n vertices and m edges. The degree of a vertex v in G is the number of edges incident with v and it is denoted by dG (v ). A graph G is regular if all its vertices have the same degree, contrarily it is irregular. The imbalance of an edge e = uv ∈ E, defined as imb(e) = |dG (u) − dG (v )|, appears implicitly in Ramsey problems with repeated degrees [12], and later in the work of Chen, Erdős, Rousseau, and Schelp [17], where 2-colorings of edges of a complete graph were considered. In [11], Albertson defined the irregularity of G as the sum of imbalances of all edges of a graph, i.e.,

∑

irr(G) =

imb(e) =

∑

|dG (u) − dG (v )|.

uv∈E(G)

e∈E(G)

For results on the Albertson irregularity index we refer the readers to [1,6,11,26,28]. To overcome certain shortcomings of Albertson’s irregularity index, recently, in [5] a new measure of the irregularity of a graph, the so-called the total irregularity of a graph, was defined as irrt (G) =

1 2

∑

|dG (u) − dG (v )| .

(u,v )∈V 2 (G)

Results on the total irregularity as well as a comparison between the irregularity irr and the total irregularity irrt of a graph were studied in [2–4,19,20,30].

*

Corresponding author. E-mail addresses: [email protected] (H. Abdo), [email protected] (D. Dimitrov), [email protected] (I. Gutman).

https://doi.org/10.1016/j.dam.2018.05.013 0166-218X/© 2018 Elsevier B.V. All rights reserved.

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Two other most frequently used graph topological indices, which measure how irregular a graph is, are the variance of degrees, firstly considered by Bell [14], and the Collatz–Sinogowitz index [18], defined as Var(G) =

n 1∑

n

d2G (vi ) −

i=1

1

(

n2

n ∑

)2 dG (vi )

,

i=1

and CS(G) = λ1 (G) − d(G), where {v1 , . . . , vn } is the vertex set of the graph G, λ1 (G) is the largest eigenvalue of the adjacency matrix of G, called the spectral radius of G, and d(G) is the average degree of G. There have been other attempts to determine how irregular a graph is [8–10,13,15,16,29], but heretofore this has not been captured by a single parameter as it was done by the irregularity measure by Albertson, the total irregularity or Bell’s irregularity. Some of the irregularity measures are also related to the first Zagreb index M1 (G) and the second Zagreb index M2 (G), one of the oldest and most investigated topological graph indices, and the so-called forgotten index, defined as follows: M1 (G) =

∑

d2G (v ) =

v∈V (G)

M2 (G) =

∑

(dG (u) + dG (v )),

uv∈E(G)

∑

dG (u)dG (v ),

and

uv∈E(G)

F(G) =

∑

d3G (v ).

v∈V (G)

More about Zagreb indices and the forgotten index can be found in [7,21,22,24,25]. A natural extension of the well-known Albertson irregularity index leads to the irregularity index σ (G), very recently introduced in [27], which is defined as

σ (G) =

∑

(dG (u) − dG (v ))2 .

uv∈E(G)

The aim of this paper is to characterize graphs with maximal σ irregularity index, and to explore the lower bounds on the maximal σ irregularity of graphs with fixed minimal and/or maximal vertex degrees. Before we continue with the results, we introduce the notation that will be used in the rest of the paper. A universal vertex is a vertex adjacent to all other vertices. The diameter of a graph G is the maximal distance between any two vertices of G. A set of vertices is said to be independent when the vertices are pairwise non-adjacent. The vertices from an independent set are independent vertices. By NG (u), we denote the set of vertices that are adjacent to a vertex u. A clique of a graph G is a complete subgraph of G. The union 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 the vertex set V = V1 ∪ V2 and the edge set E = E1 ∪ E2 . The join G = G1 + G2 of the graphs G1 and G2 is the graph union G = G1 ∪ G2 together with all the edges joining V1 and V2 . The clique–star graph KSp,q is the join graph of a clique of size p and an independent set of size q (see Fig. 3). A graph G is stepwise irregular if for any edge uv ∈ E(G), it holds that |d(u) − d(v )| ≤ 1, and it is strictly stepwise irregular if for any edge uv ∈ E(G), it holds that |d(u) − d(v )| = 1. In what follows we list the elementary properties of σ irregularity. Some of them were already presented in [23]. The proofs of these properties are straightforward and will be omitted.

• • • • • • • • •

σ (G) = irr(G) = 0 if and only if G is regular. σ (G) = irr(G) > 0 if and only if G is stepwise irregular. σ (G) = irr(G) = m if and only if G is strictly stepwise irregular. σ (G) = F (G) − 2M2 (G). If T is a tree on n vertices, then σ (T ) ≤ (n − 1)(n − 2)2 . For any graph G, σ (G) = O(n2 |E |) = O(n4 ). If a graph G is strictly stepwise irregular, then it is bipartite. For any irregular graph G, σ (G) is an even integer. If a graph G is strictly stepwise irregular, then m is even.

In the next section, we characterize general graphs with maximal σ irregularity. 2. General graphs with maximal σ irregularity In order to characterize graphs with maximal σ irregularity, we first determine the minimum number of universal vertices that such graphs must have. Please cite this article in press as: H. Abdo, et al., Graphs with maximal σ https://doi.org/10.1016/j.dam.2018.05.013.

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Fig. 1. Transformation T1 from Lemma 2.1.

Lemma 2.1. Let G be a graph with maximal σ irregularity among all graphs of order n. Then, G has at least

⌊n⌋ 4

universal vertices.

Proof. Assume that G is a graph with maximal σ irregularity whose set U of universal vertices has cardinality q < ⌊ 4n ⌋. Let U = {u1 , . . . , un−q } be the set of non-universal vertices, where d(u1 ) ≥ d(u2 ) ≥ · · · ≥ d(un−q−1 ) ≥ d(un−q ). Consider an edge ux uy , d(ux ) ≥ d(uy ), such that both ux and uy are not adjacent to u1 and ux uy is an edge whose sum of its indices (x + y) is maximal among all edges whose both end-vertices in U are not adjacent to u1 . Then replace the edge ux uy with the edge u1 ux , obtaining a graph G′ . By this transformation, denoted by T1 (Fig. 1), the number of edges remains the same, as well as the degree of ux . The contribution of the edge ux uy to σ (G) is

)2

dG (uy ) − dG (ux ) ,

(

(1)

and the contribution of the edge u1 ux to σ (G ) is ′

(dG (u1 ) + 1 − dG (ux ))2 .

(2)

The edges between uy and the universal vertices of G contribute to the change of the σ irregularity by q(−(n − 1 − d(uy ))2 + (n − d(uy ))2 ),

(3)

and the edges between u1 and the universal vertices of G contribute to the change of the σ irregularity by q(−(n − 1 − d(u1 ))2 + (n − 2 − d(u1 ))2 ).

(4)

Further, the edges between u1 and its neighbors in U, denoted by wi , i = 1, . . . , d(u1 ) − q, cause the following change of the σ irregularity d(u1 )−q

∑

(−(d(u1 ) − d(wi ))2 + (d(u1 ) + 1 − d(wi ))2 ),

(5)

i=1

while he edges between uy and its neighbors in U, denoted by vi , i = 1, . . . , d(uy ) − q, cause the following change of the σ irregularity d(uy )−q

∑

(−(d(uy ) − d(vi ))2 + (d(uy ) − 1 − d(vi ))2 ).

(6)

i=1

Clearly the expression (2) is greater than the expression (1). The sum of the expressions (3) and (4) after simplification is 2q(1 + d(u1 ) − d(uy )) which is strictly larger than 0. Every summand of the expression (5) is positive. Lastly, we consider the expression (6). If vi is adjacent to u1 , then we consider the contribution of vi in (5) and (6), and we obtain −(d(uy ) − d(vi ))2 + (d(uy ) − 1 − d(vi ))2 − (d(u1 ) − d(vi ))2 + (d(u1 ) + 1 − d(vi ))2 > 0. If vi is not adjacent to u1 , then d(vi ) ≥ d(uy ), since otherwise ux uy cannot be an edge whose sum of indices (x + y) is the maximal among all edges whose both end-vertices in U are not adjacent to u1 . For d(vi ) ≥ d(uy ) it follows that −(d(uy ) − d(vi ))2 + (d(uy ) − 1 − d(vi ))2 is positive. Thus, we have shown that the sum of the expressions (5) and (6) is positive, and consequently the sum of the expressions (1) - (6), is positive, i.e., σ (G′ ) > σ (G). We apply the above kind of replacement to all edges whose endpoints are not adjacent to u1 . Note also that u1 cannot be a universal vertex, as it would contradict the assumption that a graph with maximal σ irregularity has at most q universal vertices. Therefore, u1 is still the vertex of U of maximal degree. We denote by G1 the newly obtained graph. Consider an edge ux uy , x < y, such that (i) ux is not adjacent to u1 , (ii) uy ∈ U ∩ NG (u1 ) is adjacent to ux , and (iii) ux uy is an edge whose sum of his indices (x + y) is smallest among all edges with end-vertices in U that satisfy (i) and (ii). Then replace the edge ux uy with the edge u1 ux , obtaining a graph G′′ . By this transformation denoted by T2 (Fig. 2), the number of edges remains the same, as well as the degree of ux . The changes of the σ irregularity after this transformation can be described also here as in the previous transformation with expressions (1)–(5). The arguments, which support the fact that the sum of the expressions (1)–(5) is positive, are identical as above and will be omitted. Please cite this article in press as: H. Abdo, et al., Graphs with maximal σ https://doi.org/10.1016/j.dam.2018.05.013.

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Fig. 2. Transformation T2 from Lemma 2.1.

Fig. 3. Graphs of order 4, . . . , 15 with maximal σ irregularity index.

Hence, we can apply this second replacement to all edges between vertices of U \ NG1 (u1 ) – which only increases σ (G) – and so assume that there are none. We denote by G2 the newly obtained graph. As previously, u1 cannot become universal because of this procedure — that would in fact contradict our assumptions on G. Thus, U \ NG (u1 ) is a nonempty independent set whose cardinality we note as z > 0. We can build a new graph G⋆ with q + 1 universal vertices from G2 , by linking u1 to its z non-neighbor. This transformation, denoted by T3 , changes the degree of z + 1 vertices: the degree of the vertex u1 increases by z and the degrees of the vertices in U \ NG2 (u1 ), increase by 1. The edges between u1 and its neighbors in G2 contribute to the change of the σ irregularity by n−q−z

∑

(−(d(u1 ) − d(ui ))2 + (n − 1 − d(ui ))2 )

i=2

+ q(−(n − 1 − d(u1 ))2 + (n − 1 − d(u1 ) − z)2 ).

(7)

Considering that d(u1 ) = n − 1 − z and further expanding the expression in (7), we obtain that it is equal to n−q−z

∑

(2z(n − 1 − d(ui )) − z 2 ) − qz 2 .

(8)

i=2

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Further, considering that q + 1 ≤ d(ui ) ≤ d(u1 ), i = 2, . . . , n − q − z, it can be derivated that the expression z 2 (n − 2q − z − 1)

(9)

is a lower bound on the expression in (8). The change of the σ irregularity caused by the edges between the universal vertices and z vertices in U \ NG1 (u1 ) is zq(−(n − 1 − q)2 + (n − 1 − q − 1)2 ) = zq(−2(n − q) + 3).

(10)

Finally, the new z edges contribute to the change of the σ irregularity by z(n − q − 2)2 .

(11)

From (9)–(11), we obtain

σ (G⋆ ) − σ (G2 ) ≥ z 2 (n − 2q − z − 1) + zq(−2(n − q) + 3) + z(n − q − 2)2 = z(4 − 4n + n2 + 7q − 4nq + 3q2 − z + nz − 2qz − z 2 ).

(12)

Note that 1 ≤ z ≤ n − q − 1. The expression −z + nz − 2qz − z 2 is minimal for z = n − q − 1 and has a value of −q(n − q − 1) < 0. It follows that the expression in (12) is also minimal for z = n − q − 1 and, thus,

( ) σ (G⋆ ) − σ (G2 ) ≥ (n − q − 1) 4 − 4n + n2 + 8q − 5nq + 4q2 .

(13)

Since n > q, the expression 4 − 4n + n + 8q − 5nq + 4q is minimal when q is maximal. By the assumption, q < ⌊ ⌋. In addition, it holds that q ≤ 4n − 1 ≤ ⌊ 4n ⌋, and the lower bound of the above expression is obtain for q = 4n − 1, and thus, 2

σ (G⋆ ) − σ (G2 ) ≥

3n2 4

n 4

2

> 0.

(14)

Therefore, we have obtained a graph G⋆ with q + 1 universal vertices with σ irregularity greater than G, which is a contradiction to the assumption that a graph with maximal irregularity has at most q universal vertices. □ Observe that if we set q = Namely, we obtain that

n , 4

then the change of the σ irregularity in the above transformation is negative for n ≥ 4.

1

σ (G⋆ ) − σ (G2 ) ≥ − (n − 2)(3n − 4) < 0. 2

This indicates that the number of universal vertices q does not exceed n/4. We will now determine the graphs whose irregularity is maximum. Theorem 2.1. If a graph G has maximal σ irregularity among all graphs of order n, then G=

⎧ ⎨KS⌈ n ⌉,⌊ 3n ⌋ , 4

⎩KS

⌊ 4n ⌋,

if

4

⌈

3n 4

⌉,

n ≡ 3 (mod 4),

otherwise.

Proof. To determine the graphs with maximal σ irregularity of order at most 4 is trivial, so we assume further that n > 4. We assume that n > 4 Let G be a graph with maximum σ irregularity, let U = {vn−q+1 , vn−q+2 , . . . , vn } be the set of universal vertices and let U = {u1 , u2 , . . . , un−q } be the rest of the vertices of G. By Lemma 2.1, we have that q ≥ ⌊ 4n ⌋, and by the transformations T1 and T2 used there, it follows that a subset of vertices U, denoted further by U 1 , are adjacent all to u1 , while the rest of the vertices of U, denoted by U 2 , are adjacent only to the universal vertices. We write it as U = U 1 ∪ U 2 , with |U 1 | = n − q − z and |U 2 | = z, 0 ≤ z ≤ n − q. Next, we show that z = n − q. Assume that this is not true. Then we apply the following transformation, denoted by T3 : we delete all edges between u1 and the rest of the vertices in U 1 , obtaining the graph G∗ . After this transformation the degree of u1 decreases by n − q − z − 1 and the degrees of the rest of the vertices of U 1 decrease by 1. After deleting the edges between u1 and its n − q − z − 1 neighbor vertices in U 1 the σ irregularity decreases by n−q−z

∑ 2

n−q−z

(dG (u1 ) − dG (ui ))2 =

∑

(n − z − 1 − dG (ui ))2 .

(15)

2

The edges between u1 and the universal vertices of G contribute to the change of the σ irregularity by q(−(n − 1 − (n − z − 1))2 + (n − 1 − q)2 ) = q(−z 2 + (n − 1 − q)2 ). Please cite this article in press as: H. Abdo, et al., Graphs with maximal σ https://doi.org/10.1016/j.dam.2018.05.013.

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Finally, the edges between the neighbors of u1 in U 1 and the universal vertices of G contribute to the change of the σ irregularity by n−q−z

q

∑

(−(n − 1 − dG (ui ))2 + (n − 1 − (dG (ui ) − 1))2 )

2 n−q−z

=q

∑

(2(n − 1 − dG (ui )) + 1).

(17)

2

Observe that the edges between the neighbor vertices of u1 in U 1 do not change the σ irregularity, since all of these vertices decrease their degrees by 1. Thus, we have n−q−z

∑

σ (G∗ ) − σ (G) =

(q(2(n − 1 − dG (ui )) + 1) − (n − z − 1 − dG (ui ))2 )

2

+ q(−z 2 + (n − 1 − q)2 ).

(18)

The function 2q(n − 1/2 − dG (ui )) − (n − z − 1 − dG (ui )) is quadratic in dG (ui ) and has it maximum at dG (ui ) = n − 1 − q − z. We are interested in its global minimum, which can be achieved for the boundary values dG (ui ) = q + 1 or dG (ui ) = n − 1 − z. Indeed it global minimum of 2(−2 + n − q)q − (2 − n + q + z)2 is attained for dG (ui ) = q + 1. It follows that 2

σ (G∗ ) − σ (G) ≥ (n − q − z − 1)(2(−2 + n − q)q − (2 − n + q + z)2 ) + q(−z 2 + (n − 1 − q)2 ).

(19)

The first derivative with respect to q of the right-hand expression of the inequality (19) is always positive, and thus, the expression increases in q and it is minimal when q is minimal, i.e., q = ⌊ 4n ⌋. For q = 4n − 1, we obtain

σ (G∗ ) − σ (G) ≥

) 1 ( 21n2 z − 15n2 − 40nz 2 − 16nz + 12n + 16z 3 + 48z 2 − 16z . 16

⌉. For z = 3n − 1, the last expression is The expression)decreases in z, and it is smallest when z is maximal, i.e., z = ⌈ 3n 4 4 ( last 1 2 3n − 15n + 12 and it is positive for n > 4. Thus, we have shown that the change of the σ irregularity is positive after 4 applying the transformation T3 . The σ irregularity of KSq,n−q is q(n − q)(n − 1 − q)2 , and it is maximized for q = ⌈ 4n ⌉, if n ≡ 3 (mod 4) and for q = ⌊ 4n ⌋, otherwise. □ Corollary 2.1. For any G, σ (G) ≤ ⌊ 4n ⌋⌈ 3n ⌉ ⌈ 3n ⌉−1 4 4

(

)2

= σ (KS⌊ n ⌋,⌈ 3n ⌉ ). 4

4

In Fig. 3 all graphs of order up to 15 with maximal σ irregularity index are depicted. We would like to mention, that it is not a surprise that the graphs with maximal σ irregularity index have similar structure as the graphs with maximal Albertson irregularity index. Namely the graphs with maximal Albertson irregularity index are also clique–star graphs, but with larger size of the clique, ⌊ 3n ⌋ or ⌈ 3n ⌉ depending on the parity of n [6]. 3. Lower bounds on graphs with maximal σ irregularity In this section, we consider graphs with maximal σ irregularity and prescribed minimal or/and maximal degrees. The constructions used in the proofs in these sections are adapted from [6], where the lower bounds on graphs with maximal Albertson irregularity were considered. First, we show a lower bound for graphs with fixed maximal degree ∆. Proposition 3.1. Let G be a connected graph with n vertices with maximum degree ∆(G) = ∆, and maximal σ irregularity. Then, it holds that

σ (G) ≥

27∆3 n 256

+ nO(∆2 ).

Proof. To obtain the bound we consider the graph Q which is illustrated in Fig. 4. To simplify the calculation, we assume that ∆/4 and n/(∆ + 1) are integers. The construction of Q is as follows:

• Make a sequence of n/(∆ + 1) copies of KS ∆ , 3∆ +1 . 4

4

• Choose an edge from the first KS ∆ , 3∆ +1 graph, such that one endvertex belongs to the independent set of vertices, 4

4

and an edge from the second KS ∆ , 3∆ +1 graph, also with an endvertex form the independent set of vertices. Let denote 4

these edges by v 1∆ u13∆ 4

4

+1

4

and v12 u21 , respectively. Replace v 1∆ u13∆ 4

4

+1

and v12 u21 by edges v 1∆ u21 and v12 u13∆ 4

4

+1

. Continue this

kind of replacement between all consecutive copies of KS ∆ , 3∆ +1 . Notice that these replacements do not change the 4 4 degrees of the vertices. Please cite this article in press as: H. Abdo, et al., Graphs with maximal σ https://doi.org/10.1016/j.dam.2018.05.013.

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Fig. 4. A connected graph Q constructed from n/(∆ + 1) copies of KS ∆ , 3∆ +1 . The dashed edges are those that are removed from the corresponding 4

4

clique–star graphs.

We have

σ (Q ) =

n

∆

∆+1 4

(

3∆ 4

)( +1

3∆

)2

4

=

27∆3 n 256

+ nO(∆2 ).

□

Next, we show a lower bound for graphs with maximal σ irregularity and fixed minimal degree δ . Proposition 3.2. Let G be a connected graph with n vertices with minimal degree δ (G) = δ , and maximal σ irregularity. Then

σ (G) ≥ δ (n − δ )(n − 1 − δ ) = Ω (δ n3 ). Proof. The lower bound is attained by KSδ,n−δ whose σ irregularity is δ (n − δ )(n − 1 − δ )2 .

□

Finally, we show a lower bound for graphs with maximal σ irregularity and fixed maximal and minimal degrees. Proposition 3.3. Let G be a connected graph with n vertices with minimal degree δ (G) = δ , maximal degree ∆(G) = ∆, and maximal σ irregularity. Then, it holds

σ (G) >

δ (∆ − δ )3 n. ∆+1

Proof. To obtain the bound we consider the graph R, which is constructed in the same way as the graph Q in Fig. 4, with the only difference that R is built of n/(∆ + 1) copies of KSδ,∆−δ+1 . To simplify the calculation, we assume that n/(∆ + 1) is integer. We have n σ (R) = δ (∆ − δ + 1)(∆ − δ )2 . □ ∆+1 We believe that the graphs used in the proofs of the results in this section lead to sharp bounds, or very close to them. To prove that these bounds are sharp or eventually to find the sharp bounds in the above settings is still open. Acknowledgments The authors would like to thank the anonymous reviewers for the meaningful remarks, which helped to improve the quality of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8]

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irregularity, Discrete Applied Mathematics (2018),