On some properties of graph irregularity indices with a particular

Applied Mathematics and Computation 344–345 (2019) 107–115

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

On some properties of graph irregularity indices with a particular regard to the σ -index Tamás Réti Obuda University, Népszínház str. 8, Budapest H-1081, Hungary

a r t i c l e

i n f o

Keywords: Graph irregularity Complete split graph Stepwise irregular graph

a b s t r a c t Some well-known graph irregularity indices of a connected graph G are investigated. Our study is focused mainly on the comparison of the Bell’s degree-variance (Var(G)) and the Collatz–Sinogowitz irregularity index (CS(G)) with the recently introduced σ (G) irregularity index. It is a degree-based topological invariant calculated as σ (G ) = F (G ) − 2M2 (G ) where  M2 (G) is the second Zagreb index, F (G ) = d3 (v ), and d (v ) is the degree of the vertex v in G. By introducing the notion of the complete split-like graphs representing a broad subclass of bidegreed connected graphs, it is shown that for these graphs the equality σ (G ) = n2Var (G ) holds. © 2018 Elsevier Inc. All rights reserved.

1. Introduction All graphs considered in this study are finite simple connected graphs. For a graph G with n vertices and m edges, V(G) and E(G) denote the set of vertices and edges, respectively. To avoid trivialities, we always assume that n ≥ 3. The degree of a vertex v is the number of edges incident to v, and denoted by d (v ) where d (v ) ≥ 1. Let  = (G ) and δ = δ (G ) be the maximum and the minimum degrees, respectively, of vertices of G. An edge of G connecting vertices u and v is denoted by uv. For two different vertices u and w, the distance d (u, w ) between u and w is the number of edges in a shortest path connecting them. The diameter of a connected graph G denoted by diam(G) is the maximum distance between any two vertices of G. Using the standard terminology [10,16,18,32], let A = A(G ) be the adjacency matrix of G. The spectrum of a graph G denoted by Spec(G) is the set of graph eigenvalues of the adjacency matrix. For a graph G, we denote by λ(G) the largest eigenvalue of A(G) and call it the spectral radius of G. A graph is R-regular if all its vertices have the same degree R, contrarily it is irregular. A universal vertex of a graph is a vertex adjacent to all other vertex. A connected graph G is said to be bidegreed with degrees  and δ ( > δ if at least one vertex of G has degree  and at least one vertex has degree δ , and if no vertex of G has a degree different from  or δ . A connected bidegreed bipartite graph G(, δ ) is called semiregular if each vertex in the same part of bipartition has the same degree. Based on the considerations outlined in [3,21], a graph G is stepwise irregular if for any edge uv, it holds that |d (u ) − d (v )| ≤ 1 , and it is strictly stepwise irregular if for any edge uv, it holds that |d (u ) − d (v )| = 1 . The traditional complete bipartite graphs Kp, q with q = p + 1 represent a subset of strictly stepwise irregular graphs.

E-mail address: [email protected] https://doi.org/10.1016/j.amc.2018.10.010 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

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An n-vertex complete split graph KS(n, q) is a connected bidegreed graph consisting of an independent set of n − q vertices and a clique of q vertices, such that each vertex of the independent set is adjacent to each vertex of the clique [5,7,8,32]. 2. Preliminary considerations By definition, a topological invariant IT(G) of a graph G is called an irregularity index, if IT(G) ≥ 0 and IT (G ) = 0 if and only if graph G is a regular graph. There exist several degree-based and eigenvalue-based graph irregularty indices [1,2,6,9,14,20,23,28,29]. The irregularity of a connected n-vertex and m-edge graph can be characterized by the Collatz-Sinogowitz irregularity index [14]

CS(G ) = λ(G ) −

2m n

(1)

where λ is the spectral radius of the graph G. Due to the simple computation, the majority of irregularity indices are degree-based. In the mathematical chemistry the most frequently used graph irregularity indices are the variance of degree VAR(G) introduced by Bell [9]

V ar (G ) =

  2 m 2 1  1 2 1 2 d (u ) − 2 d (u ) 2 = d (u ) − n n n n u∈V

u∈V

(2)

u∈V

and the Albertson’s irregularity index [6]

AL(G ) =



|d ( u ) − d ( v )|.

(3)

uv∈E

Let M1 (G) and M2 (G) be the first and second Zagreb indices of a graph G defined as [19,22,27,33,34,36,38] M1 (G ) =  2 u∈V d (u ) and M2 (G ) = uv∈E d (u )d (v ) and denote by F(G) the so-called forgotten topological index introduced in [17] and defined as 

F (G ) =



d 3 ( u ).

u∈V

Using the degree-based topological invariants M1 (G), M2 (G) and F(G) several versions of irregularity indices can be generated. By the natural extension of the Albertson irregularity index the so-called σ -index has been introduced [24]

σ (G ) =





d (u ) − d (v ) 2 .

(4)

uv∈E

Some elementary properties of σ (G) irregularity index are listed in [3]: • • • • • •

σ (G ) = CS(G ) = V ar (G ) = AL(G ) = 0 if and only if G is regular. σ ( G ) = F ( G ) − 2M2 ( G ). If G is a connected graph with n vertices and m edges, then σ (G ) ≤ m(n − 2 )2 with equality if and only if G is a star. If T is a tree graph on n-vertices, then σ (T ) ≤ (n − 1 )(n − 2 )2 with equality if and only if T is a star. σ (G ) = AL(G ) = m if and only if G is a strictly stepwise irregular graph. For any irregular graph G, σ (G) is an even integer.

Recently, it has been proved that connected graphs with maximal σ -irregularity are represented by a particular set of complete split graphs [3]. Starting with the Zagreb indices several irregularity indices of new type can be established. Such irregularity indices are [28,29]



IRM1 (G ) =

 IRM2 (G ) =



 2m M1 ( G ) − = V ar (G ) n n 2m M2 ( G ) − . n n

M1 ( G ) 2m −1 + n n

(5)

(6)

Using the inequality reported in [13] a simple measure of the irregularity of a graph can be constructed. It is defined as

IRC (G ) = F (G ) −

2m M1 ( G ) ≥ 0 . n

(7)

Additionally, we introduce a novel irregularity index IRV(G) which is considered as a modified version of Var(G) index

IRV (G ) = n2V ar (G ).

(8)

The irregularity index IRV(G) will play a central role in the following investigations. It is easy to see that if graphs GA and GB have identical degree sequences then equalities IRM1 (GA ) = IRM1 (GB ), IRC (GA ) = IRC (GB ) and IRV (GA ) = IRV (GB ) hold.

T. Réti / Applied Mathematics and Computation 344–345 (2019) 107–115

Lemma 1 ([37]). Let G be a connected graph with a spectral radius λ. Then λ2 (G ) ≥ a regular connected graph or a semiregular bipartite graph.

M1 ( G ) n

109

with equality if and only if G is either

Proposition 1. Let G be a connected graph with n vertices, m edges and a spectral radius λ. Then

IRV (G ) = n2V ar (G ) ≤ (2mn + λn2 )(CS(G ) and equality holds if G is regular or semiregular. Proof. According to Lemma 1 if G is a connected graph then the inequality λ2 (G ) ≥

n2V ar (G ) = n2

 M1 ( G ) n

−

 2 m 2  n

≤ n2



Moreover, if G is semiregular, then λ2 (G ) =

λ+

is valid. It follo ws that

 2m  2m  2m  = n2 λ + CS(G ). λ− n n n

M1 ( G ) n ,

consequently





M1 ( G ) n

n2V ar (G ) = (2mn + λn2 )CS(G ) = 2nm + n2

M1 ( G )  CS(G ). n



In what follows the relations between that above irregularity indices are investigated with a particular regard to σ (G), IRV(G) and CS(G) indices. Our further investigations are closely related to the connected graphs having exactly two main eigenvalues [12,25,30,31]. An eigenvalue ρ of the adjacency matrix is said to be a main eigenvalue of a graph G, if the eigenspace (ρ ) is not orthogonal to the all-one vector j whose all coordinates are equal to one [16]. The spectral radius λ of a graph G is always a main eigenvalue, and a graph G has exactly one mai n eigenvalue if and only if G is regular. We will focus on the case where G is a bidegreed graph possessing exactly two main eigenvalues denoted by λ and μ where λ > μ. As it is known, the complete split graphs, the semiregular graphs, the wheel graphs and the friendships graphs belong to the family of bidegreed graphs having 2 main eigenvalues [5,7,8,11,31]. 3. Some properties of complete split graphs For complete split graphs the Collatz–Sinogowitz irregularity index and the σ index can be simply computed. Lemma 2 ([26]). Let G be a connected n-vertex and m-edge graph with the spectral radius λ(G). Then

λ (G ) ≤

δ−1+



( δ + 1 )2 + 4 ( 2m − δ n ) 2

.

Equality holds if and only if G is regular or a bidegreed graph in which each vertex is of degree either δ or n − 1. As a consequence of Lemma 1 the following corollary yields: Corollary 1. In the case of complete split graphs,  = n − 1, δ = q hold, this implies that for their edge number

m(KS(n, q )) = δ (n − δ ) + δ (δ − 1 )/2 = q(n − q ) + q(q − 1 )/2 consequently,

2m(KS(n, q )) = (2n − 1 )δ − δ 2 = (2n − 1 )q − q2 . Using Lemma 1 one obtains for the corresponding spectral radius [7,8]:

   1  1 δ − 1 + (δ + 1 )2 + 4(2m − δ n ) = q − 1 + 4qn − 3q2 − 2q + 1 . 2 2

λ(KS(n, q )) =

Based on the above equation, the Collatz–Sinogowitz irregularity index of a complete split graph can be calculated as

CS(KS(n, q )) =

 1 1 (q − 1 + 4qn − 3q2 − 2q + 1) − 2q + (q + q2 ). 2 n

It is interesting to note that

lim

n→∞

 2m(KS(n, q ))  n



= lim 2q − n→∞

q + q2  = 2q n

this implies that the inequality n − 2 ≥ q ≥ m/n ≥ 1 holds where x denotes the least integer no less then x. Lemma 3 ([8]). Let KS(n, q) be a complete split graph. Then KS(n, q) has two main eigenvalues λ and μ < λ. These are:

λ= μ=

q−1+

q−1−



4qn − 3q2 − 2q + 1 2



4qn − 3q2 − 2q + 1 . 2

(9)

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In the formulas above, λ is the spectral radius, and μ is the smallest eigenvalue. Moreover it follows that for a complete split graph

λ + μ = q − 1 ≥ 0. Lemma 4 ([30]). Let G be a connected graph with mean degree 2m/n, and just two main eigenvalues, λ and μ < λ, where λ is the spectral radius of G. Then

V ar (G ) =

   2 m 2   2m 2m  2m 1  2 d (u ) − = λ− − μ = CS(G ) −μ n n n n n

(10)

u∈V (G )

Lemma 5. Let KS(n, q) be a complete split graph. Then

V ar (KS(n, q )) =

 2 m 2  2m  2m 1  2 d (u ) − = q (n − q ) − −q+1 . n n n n

(11)

u∈V (G )

Proof. Because a complete split graph has exactly two main eigenvalues, by performing simple computations the result follows from Lemma 2 and Lemma 4.  Lemma 6. Let KS(n, q) be a complete split graph with m edges. Because m(KS(n, q )) = q(q − n ) + q(q − 1 )/2 one obtains that

σ (KS(n, q )) =



 q (q − 1 )  (d (u ) − d (v ))2 = m − (n − 1 − q )2 = q(n − q )(n − 1 − q )2 . 2

uv∈E

Lemma 7. Let KS(n, q) be a complete split graph. Then by Lemma 5 we get



n2V ar (KS(n, q )) = n2 q(n − q ) −

 2m  2m − q + 1 = q(n − q )n2 − 2m(2m − qn + n ). n n

From above Lemmas one obtains the following proposition. Proposition 2. Let KS(n, q) be a n-vertex and m-edge complete split graph. Then

M1 (KS(n, q )) = (n − q )q2 + q(n − 1 )2 and

n2V ar (KS(n, q )) = n2

 M1 (KS(n, q ))

− n 2 2 −[(2n − 1 )q − q ]

 2 m 2 n

= nM1 (KS(n, q )) − 4m2 = n[(n − q )q2 + q(n − 1 )2 ]

consequently,

σ (KS(n, q )) = n2V ar (KS(n, q )) = q(n − q )(q − 1 − q )2 .

(12)

Lemma 8 ([5]). Let H be an n-vertex and m-edge connected graph with exactly two main (distinct) eigenvalues λ(H) and μ(H). Then

μ (H ) =

M1 (H ) − 2mλ(H ) . 2 m − nλ ( H )

(13)

Using Lemma 4 and Lemma 8 the following proposition can be obtained. Proposition 3. Let H be a connected n-vertex and m-edge graph with exactly two main eigenvalues λ > μ, where λ is the spectral radius of H. Then

n2V ar (H ) = n2



λ−

 2m  2m − μ = (2mn − μn2 )CS(H ) n n

where μ is computed by Eq. (13). 4. Well-stabilized graphs Based on the concept outlined in [39], a connected graph is said to be a well-stabilized graph, if any two non-adjacent vertices have equal degrees. It is obvious that connected regular graphs are well-stabilized. Among irregular bidegreed graphs, the star graphs, the complete bipartite graphs, the bidegreed complete multipartite graphs, the friendship graphs FRk constructed by joining k copies of the cycle C3 with a common vertex, the wheel graphs Wn with n ≥ 5 vertices can be considered as particular types of well-stabilized graphs. Small well-stabilized graphs are illustrated in Fig. 1. Example 1. Consider the complete bipartite graph Kp, q . Because Kp, q is a well-stabilized graph then

√

σ (Kp,q ) = n2V ar (Kp,q ) = (2mn + n2 m )CS(Kp,q ).

T. Réti / Applied Mathematics and Computation 344–345 (2019) 107–115

111

Fig. 1. Small well-stabilized graphs with 6 and 7 vertices.

Proof. For the complete bipartite graph Kp, q with n vertices and m edges we have n = p + q, m = pq, M1 (K p,q ) = nm and √ √ λ(Kp,q ) = pq = m. Moreover,

σ (Kp,q ) =



(d (u ) − d (v ))2 = m( p − q )2

uv∈E

and

V ar (K p,q ) =

√ M1 (K p,q )  2m 2 4m2 2m  2m √ − =m− 2 = m+ m− . n n n n n

This implies that

n2V ar (K p,q ) = n2 m − 4m2 = m(n2 − 4m ) = m(( p + q )2 − 4 pq ) = m( p − q )2 = σ (K p,q ). Because

CS(K p,q ) = λ −

2m √ 2m = m− n n

one obtains that

n2V ar (K p,q ) = n2

√

m+

√ 2m  2m √ m− = (2mn + n2 m )CS(K p,q ). n n

As a particular case, for a star graph K1,n−1 , m = n − 1, λ(K1,n−1 ) =

√ n − 1 hold. This implies that

 σ (K1,n−1 ) = n2V ar (K1,n−1 ) = (2n(n − 1 ) + n2 n − 1 )CS(K1,n−1 ). 

Remark 1. It should be noted the complete bipartite graphs Kp, q with p = q are semiregular and well-stabilized graphs, but there exist semiregular graphs not belonging to the family of well-stabilized graphs. For example, the 12-vertex subdivision graph of complete graph K4 is a semiregular and strictly stepwise irregular graph, but not a well-stabilized graph. Example 2. Consider the friendship graph FRk with n = 2k + 1 vertices and m = 3k edges. Graphs FRk are well-stabilized bidegreed graphs having exactly two main eigenvalues λk and μk , [4]:

   λk = 1 + 1 + 8k /2

and

   μk = 1 − 1 + 8 k / 2 .

For the friendship graphs the following identity holds:

σ (F Rk ) = n2V ar (F Rk ) = (2mn − μk n2 )CS(F Rk ). Proof. It is easy to see that

σ ( F Rk ) =



(d (u ) − d (v ))2 = 8k(k − 1 )2 .

uv∈E

On the other hand, because M1 (F Rk ) = 8k + 4k2 this implies that

n2V ar (F Rk ) = nM1 (F Rk ) − 4m2 = (2k + 1 )(8k + 4k2 ) − 4(3k )2 = 8k(k − 1 )2 . Moreover, by Lemma 4 one obtains that

n2V ar (F Rk ) = n2 where



λk −

 2m  2m − μk = (2mn − μk n2 )CS(F Rk ) n n

√ 2m 6k 1 + 1 + 8k CS(F Rk ) = λk − = − . n 2 2k + 1



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Fig. 2. Complete split-like graphs KSL(5, 1, 2), KSL(5, 1, 3) and KSL(6, 2, 3) with q = 1 and 2 universal vertices.

Fig. 3. Generalized windmill graphs with 11 vertices.

5. Complete split-like graphs There exists a broad subclass of well-stabilized graphs which can be considered as the generalized versions of traditional complete split graphs. A connected n-vertex bidegreed graph is said to be a complete split-like graph, denoted by KSL(n, q, δ ), if it has q ≥ 1 universal vertices with degree n − 1 and n − q vertices with degree δ . It is easy to see that for the edge number m of a complete split-like graph 2m = (n − 1 )q + (n − q )δ holds. Complete split-like graphs form a subset of wellstabilized graphs, moreover they are considered as possible extended versions of traditional complete split graphs. This is due to the fact, that KS(n, q ) = KSL(n, q, δ ) if δ = q holds. As can be observed, if G is a complete split-like graph then the equality diam(G ) = 2 is fulfilled. As an example, in Fig. 2 small complete split-like graphs with 5 and 6 vertices are depicted. In a particular case, if q = 1 holds, a complete split-like graph is said to be a generalized windmill graph. It is denoted by KSL(n, 1, δ ). It follows that the star graphs K1,n−1 on n ≥ 3 vertices, the wheel graphs Wn on n ≥ 5 vertices, and the classical windmill graphs Wd(k, p) with (k − 1 ) p + 1 vertices and pk(k − 1 )/2 edges defined for k ≥ 2 and p ≥ 2 positive integers, form the subsets of generalized windmill graphs. (As it is known, the windmill graph Wd(k, p) can be obtained by joining p copies of the complete graph Kk with a common vertex). Various types of generalized windmill graphs can be constructed using the concept of cone graph generations [35]. A cone over a graph G is obtained by adding a vertex to G that is adjacent to all vertices of G. It is easy to see, that by adding a vertex (a universal vertex) to a regular graph GR a generalized windmill graph is obtained. As an example, generalized windmill graphs KSL(11, 1, 3) and KSL(11, 1, 4) constructed from 2- and 3-regular graphs are depicted in Fig. 3. It should be emphasized that the graphs in Figs. 2 and 3 are not complete split graphs, but they are complete split-like graphs. Based on following considerations it is easy to see that the Collatz-Sinogowitz irregularity index of complete split-like graphs can be simply computed without knowing the corresponding adjacency matrix. Because for n-vertex and m-edge complete split-like graphs the identity 2m = (n − 1 )q + (n − q )δ holds, from Lemma 1 it follows that their corresponding spectral radii are

λ(KSL(n, q, δ )) =

   1  1 δ − 1 + (δ + 1 )2 + 4(2m − δ n ) = δ − 1 + (δ + 1 )2 + 4[(n − 1 )q + (n − q )δ − δ n] . 2 2 (14)

Consequently, the Collatz–Sinogowitz irregularity index of KSL(n, q, δ ) graphs can be calculated as

CS(KSL(n, q, δ )) = λ(KSL(n, q, δ )) −

( n − 1 )q + ( n − q )δ n

.

6. A novel sharp upper bound for σ index In what follows it will be shown that starting with the complete split-like graphs a sharp upper bound can be established for the σ irregularity index.

T. Réti / Applied Mathematics and Computation 344–345 (2019) 107–115

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Lemma 9 ([39]). If G is a connected graph with n vertices and m edges, then

F ≤ 2M2 (G ) + nM1 (G ) − 4m2

(15)

with equality if G is a well-stabilized graph. Proposition 4. Let G be a connected graph with n vertices and m edges. Then

σ (G ) ≤ IRV (G ) = n2V ar (G )

(16)

where equality holds if G is a regular or a well-stabilized graph. Proof. Based on Eq. (15) one obtains that

σ (G ) = F (G ) − 2M2 (G ) ≤ nM1 (G ) − 4m2 = n2 From Lemma 9 the result follows.

 M1 ( G ) n

−

4m2  = n2V ar (G ). n2



Corollary 2. If well-stabilized graphs GA and GB possess identical degree sequences then the equality σ (GA ) = σ (GB ) holds. Example 3. Consider the 7-vertex and 12-edge generalized windmill graphs WS4 and WS5 depicted in Fig. 1. The nonisomorphic graphs WS4 and WS5 are characterized by the following spectra:



√ √ 7, 1, 1, −1, −1, 1 − 7, −2



√ √ 7, 2, −1, −1, −1, −1, 1 − 7

Spec(W S4 ) = 1 + Spec(W S5 ) = 1 +

and both of them possess two identical main eigenvalues λ = 1 + obtains

μ=

√ √ 7 and μ = 1 − 7. Because m = 12 and M1 = 90, one

√ √ M1 − 2mλ 90 − 24λ 90 − 24(1 + 7 ) = = = 1 − 7 = −1.645751. √ 2 m − nλ 24 − 7λ 24 − 7(1 + 7 )

It is worth noting √ that for the wheel graph WS4 the least eigenvalue is −2, while the least eigenvalue of graph WS5 is equal to μ = 1 − 7. 7. Final remarks An important observation is that the spectrum does not characterize the combinatorial structure of split-like graphs. The generalized windmill graph WS4 depicted in Fig. 1 is cospectral with the 7-vertex graph No. 12–542 included in [15]. This graph has no universal vertices, its degree set is {2,3,4}. There exist generalized integral windmill graphs with two main eigenvalues. A simple example is the 7-vertex friendship graph FR3 containing 3 triangles with a common vertex. It has a spectrum



Spec(F R3 ) = 3, 1, 1, −1, −1, −1, −2



and two main eigenvalues: λ = 3 and μ = −2. Another example is the 11-vertex generalized windmill graph reported in [15]. Graph G11 depicted in Fig. 4 is the cone over the 3-regular Petersen graph with spectrum





Spec(G11 ) = 51 , 15 , −25 . It is easy to check that G11 has two main eigenvalues: λ = 5 and μ = −2. It should be emphasized that G11 does not belong to the family of traditional complete split graphs. Based on the previous considerations the following conjecture can be formulated: Conjecture 1. Let KSL(n, q, δ ) be a complete split-like graph with a spectral radius λ. Then G has exactly two main eigenvalues λ and μ where μ < λ. If the conjecture 1 is true then from Lemma 7 it follows that for a complete split-like graph

   4m2  2m  2m σ (KSL(n, q, δ )) = n2V ar (KSL(n, q, δ )) = n M1 (KSL(n, q, δ )) − = n2 λ − −μ n

= (2nm − μn2 )CS(KSL(n, q, δ )) and according to Eq.(14) and Lemma 8 we get

λ(KSL(n, q, δ )) =

  1 δ − 1 (δ + 1 )2 + 4((n − 1 )q + (n − q )δ − δ n )) 2

n

n

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T. Réti / Applied Mathematics and Computation 344–345 (2019) 107–115

Fig. 4. Generalized integral windmill graph G11 with two main eigenvalues.

and

μ(KSL(n, q, δ )) =

M1 (KSL(n, q, δ )) − 2mλ . 2 m − nλ

Moreover, if the conjecture 1 is true then from the above formula it follows that

  2m  2m σ (KSL(n, q, δ )) = n2V ar (KSL(n, q, δ )) = n2 λ − −μ n n    2 M1 (KSL (n, q, δ )) − 2mλ = 2nm − n CS(KSL(n, q, δ )). 2 m − nλ References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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