Measures of irregularity of graphs

Measures of irregularity of graphs

Joelma Ananias de Oliveira^I; Carla Silva Oliveira^II; Claudia Justel^III; Nair Maria Maia de Abreu^IV,*

^IUniversidade Federal de Mato Grosso, Brasil. E-mail: [e-mail]

IIEscola Nacional de Ciências Estatísticas, Rio de Janeiro, Brasil. E-mail: [e-mail]

^IIIInstituto Militar de Engenharia, Rio de Janeiro, Brasil. E-mail: [e-mail]

IVUniversidade Federal do Rio de Janeiro, Brasil. E-mail: [e-mail]

ABSTRACT

A graph is regular if every vertex is of the same degree. Otherwise, it is an irregular graph. Although there is a vast literature devoted to regular graphs, only a few papers approach the irregular ones. We have found four distinct graph invariants used to measure the irregularity of a graph. All of them are determined through either the average or the variance of the vertex degrees. Among them there is the index of the graph, a spectral parameter, which is given as a function of the maximum eigenvalue of its adjacency matrix. In this paper, we survey these invariants with highlight to their respective properties, especially those relative to extremal graphs. Finally, we determine the maximum values of those measures and characterize their extremal graphs in some special classes.

Keywords: index of a graph, irregularity measure, extremal graphs.

1 INTRODUCTION

Let G be an undirected graph with n vertices and m edges without loops and multiple edges. We denote (n) and (n), the respective sets of all the graphs (connected or not) and all the connected graphs with n vertices and, G(n,m) and H (n,m), the respective sets of the graphs (connected or not) and all the connected graphs with n vertices and m edges. The degree of a vertex v_i, denoted d(v_i), is the number of the incident edges in v_i.Since i = 1,n v_i= 2m, the i=1,n average of vertex degrees can be given as (G) = . A graph is k-regular or, simply regular, if n every degree is equal to k. Otherwise, the graph is said to be an irregular graph.

The eigenvalues of a graph are the eigenvalues of its adjacency matrix. They are also respectively referred as the A-eigenvalue, denoted λ1 > λ2 > ... > λn. The index of a graph is the maximal eigenvalue of A(G),that is, λ1(G). The join G₁ ∨ G₂ of (disjoint) graphs G₁ and G₂ is the graph obtained from G₁ and G₂ by joining each vertex of G₁ with each vertex of G₂.

Given a graph G, what is the minimal number of edges needed to add to G to get H, a regular supergraph of G? In this case, which pairs of non adjacent vertices in G should become adjacent in H? Questions such as these are interesting and in order to try to answer them it seems useful to know how near or distant any graph is from a regular one. We define as the measure of irregularity of G, every function F : G , where G is the set of all graphs and is the set of all real number, such that ∀G ∈ , G is a regular graph if and only if F(G) = 0.

In this paper, we survey all known parameters used as measures of irregularity as well as their respective properties, mainly those relative to upper and lower bounds and their respective extremal graphs. Also, we characterize their extremal graphs among all the split complete graphs and among all path complete graphs with a particular characteristic.

2 KNOWN MEASURES OF IRREGULARITY

In 1957, Collatz & Sinogowitz [8] proved that the index of a graph is greater than or equal to the average of vertex degrees and the equality holds if and only if the graph is regular. From this fact, they introduced ε(G) =λ₁(G) - (G) to measure the irregularity of a graph, which we call here, the spectral measure. Those authors also proved that, for n <5, the maximum value to ε(G) is -2 + and the maximal is atstained for the star S_n. Based on this, they conjectured that S_nis the most irregular graph among all graphs with n vertices relative to ε(G). In 1988, this conjecture was refuted by Cvectković & Rowlinson [9].

Some years after, Bell [4] suggested making the variance of the degrees of the vertices of G,

as a measure of the irregularity of G, and we refer to it as the variance measure. Bell characterized the most irregular graphs in the classes (n), (n,m) and (n,m) concerning the both measures ε(G) and σ(G). In the same paper, he obtained upper and lower bounds of σ(G) as functions of n and m.

Albertson [2] defined the imbalance of an edge (v_i , v_j) by imb _{vi vj} =|d(v_i) - d(v_j)| and he used it to introduce irr (G) = (v_i, v_j) ∈E ^imbv_i v_j as a measure of the irregularity of G and, in the same paper, irr(G) < is proven. In 2000, Hansen et al. [10] presented a bound to irr(G) as a function of n and m and determined extremal graphs for it. Here, we refer this invariant as the imbalance measure.

More recently, Nikiforov [13] introduced s(G) = v_i∈V (G) |d (v_i) - | as a new measure of the irregularity of a graph. Also, he showed several inequalities with respect to s(G), ε(G) and σ(G). The measure of Nikiforov will be named by the degree deviation measure.

It is easy to see that ε(G), σ(G) and s(G) can be taken by F(G),since ε(G) = σ(G) = s(G) = 0 if and only if G is a regular graph. Although irr (G) is considered in the literature as an irregularity measure, it can be null even when G is an irregular graph. In this case, irr(G) = 0if and only if G is either regular or a disconnected graph with all the regular components. Figure 1 illustrates this fact.

3 EXTREMAL IRREGULARITY IN GENERAL CLASSES OF GRAPHS

This section is devoted to define the families of graphs where some elements are extremal for ε(G), σ(G) and irr(G) where G is a graph belonging to one of the most general classes such as (n), (n), (n,m) and (n,m). Then, theorems are presented to show the conditions under which those graphs are extremal for the measures studied. The section ends with a table summarizing these results.

Definition 3.1. [1] A graph with n vertices and m edges is a quasi-complete graph, denoted QC(n,m), when it is obtained as follows:

(i) its vertices are labeled 1,...,n;

(ii) the integers d and t, 2 < d, 0 < t < d, are defined such that m = + t;

(iii) the vertices labeled 1,...,d form the maximal clique;

(iv) the vertex d + 1 is adjacent to t vertices labeled 1,...,t and the n - (d + 1) remaining vertices are isolated.

Figure 2 displays some examples of quasi-complete graphs.

The first result, due to Bell [4], gives the maximum value of the spectral measure, ε(G),where G is a graph (not necessarily connected) with n vertices. Moreover, it shows that the extremal graphs to ε(G) are the quasi-complete graphs. Proposition 3.2, also due to Bell, proves that among all graphs with n vertices and m edges, no necessarily connected, the quasi-complete graph QC(n,m) is the only one that satisfies the maximal value of the spectral measure.

Proposition 3.1. [4] Given n, write m = and m^' =. Then, we have

max{ε(G) : G ∈(n)} =

This maximum is attained uniquely by QC(n, m) if n is odd, and by QC(n, m) and QC(n, m^') (only) if n is even.

For example, if n = 6, max{ε(G) : G ∈(6)} = 1 and G is either QC(6,3) = K₃ ∪3K₁ or QC(6,6) = K₄ ∪2K₁.

Proposition 3.2. [4] Let n and m be given with m < Then, max {ε(G) : G ∈(n, m)} is attained uniquely by QC(n, m).

The following result gives the maximum value of the variance measure, σ(G), when G is a graph with n vertices, and it proves that the maximum is attained where G is a quasi-complete graph.

Proposition 3.3. [4] Given n, write r = and m = . Then, we have max{σ(G) : G ∈ (n)} =(r -1)²(n - r), and this maximum is attained by QC(n,m).

For example, if n = 7, max{σ(G) : G ∈(7)} = 3,26 and G is QC(7,10) = K₅ ∪2K₁.

For specific values of n and m, the quasi-star graphs, defined as follows, give the extremal graph to the variance measure in the class of the connected graphs with n vertices.

Definition 3.2. [1] A quasi-star, denoted QS(n,m), is a graph with n vertices and m edges when it is obtained as follows:

(i) its vertices are labeled 1,..., n;

(ii) the integers d and t 2 < d, 0 < t < d, are defined such that m = - - t;

(iii) the first n - d - 1 vertices are adjacent to all others vertices of the graph;

(iv) the vertex n - d is adjacent to vertices labeled 1,..., n - t.

The Figure 3 shows quasi-star graphs with 5 and 6 vertices.

The next result is an immediate consequence of Proposition 3.4 and it was presented in [4] as its corollary. Proposition 3.5 shows that among all graphs with n vertices and m edges, the extremal ones relative to the variance measure are quasi-stars QS(n,m), when they are sparse graphs; otherwise, the extremal graphs are quasi-complete QC(n,m).

Proposition 3.4. [4] Given n, write r =and m = - . Then, we have max{σ(G) : G ∈(n)} = (r -1)²(n - r), and this maximum is attained by QS(n, m).

For example, if n =7, max{σ(G) : G ∈(7)} = 3,26 and G is QS(7,11) = K₁ ∨K_1,5.

Proposition 3.5. [4 ] Let n and m be given, with m < . Then, the max{σ(G) : G ∈(n,m)} is attained by QC(n, m) if m > + , and by QS(n, m) if m < -

For example, if n = 6 and m = 11, max{σ(G) : G ∈(6,11)} = 1,55 and G is QC(6,11) = K₁ ∨(K₄ ∪K₁). If n = 6 and m = 4, max{σ(G) : G ∈(6,4)} = 1,55 and G is QS(6,4) = S₅ ∪K₁.

Definition 3.3. [7 ] Let 0 < k <n - 3 and let H_n,k be a connected graph with n + k edges. H_n,k is obtained as follows:

(i) label the vertices from 1 to n;

(ii) add n - 1 edges from the vertex labeled 1 to 2,..., n(a star S_n with center in 1);

(iii) add k edges from the vertex 2 to each vertex 3, ..., k +3.

The graphs H_n,kresult from the following join H_n,k = K₁ ∨(K_1,k+1 ∪_n-k-3). Similar to them, Definition 3.4 introduces the graphs G_n,k. For a large enough n, the first graphs satisfy the maximum value to the spectral measure among all connected graphs with n vertices and n + k edges; the second graphs satisfy the maximum values of the variance measure among all non sparse connected graphs with n vertices. Some examples of graphs H_n,kare shown in Figure 4.

Proposition 3.6. [4 ] For fixed k > 3, and n sufficiently large (depending on k), max{ε(G) : G ∈(n,n +k)} is attained uniquely by H_n,k.

Definition 3.4. [9 ] For k ∈ such that 0 < k < , G_n,k results of the join of K₁and QC(n -1, k), that is, G_n,k = K₁ ∨QC(n -1, k).

See some examples of graphs G_n,kin the Figure 5.

Proposition 3.7. [4 ] If n and m = n + k are such that n -1 < m <, then max{σ(G) : G ∈(n, m)} is attained by G_n,k if m > + n - 1, and by QS(n, m) if m < .

Definition 3.5. [3 ] Given n and q integers such that 0 < q < n, a pineapple graph, denoted by PA(n,q), is a graph with n vertices consisting of a clique on q vertices and a stable set on the remaining n - q vertices in which each vertex of the stable set is adjacent to a unique and the same vertex of the clique.

Aouchiche et al., in [3], conjectured that, among all connected graphs with n vertices, the maximal relative to the spectral measure are the pineapple graphs such that the size of the maximal clique is one or two unities more than the size of the maximal stable set.

Conjecture 3.1. [3 ] The most irregular graph to ε(G), where G is a connected graph on n (n > 10) vertices, is a pineapple PA(n,q)) in which the clique size q is equal to + 1.

Finally, it follows the definition of the family of graphs which are maximal to the imbalance measure.

Definition 3.6. [3 ] Given n, k and t integers such that t <k <n, a fanned complete split graph, denoted by FCS(n, k, t), is a graph with n vertices obtained from a complete split graph CS(n, k) by connecting a vertex from the stable set by edges to t other vertices of the stable set.

Theorem 3.1. [10 ] For any graph G with n vertices, m edges and irregularity irr (G),

irr(G) <k(n - k)(n - k - 1) + t (t - 2k - 1)

where

and

t = m - (n - k)k - k(k - 1)/2.

Moreover, this value is attained if and only if G is a fanned complete split graph.

The Figure 6 shows the extremal fanned complete split graph with 8 vertices and 15 edges for imbalance measure, where irr (FCS(8, 2, 2)) = 54.

Table 1 shows, in each line of the third column, the extremal graph among all graphs in the class on the correspondent line of the second column. Each extremal graph is relative to the irregularity measure displayed in the same line of the first column. The last column gives the references that support those results.

4 EXTREMAL IRREGULARITY IN PC_n,p,1AND SC(n, k) GRAPHS

The results presented in this section come originally from [14] and approach the following connected graphs with order n: the graphs PC_n,p,₁ = which are special case of path complete graphs (see Definition 4.1, [5], [12]) and the complete split graphs SC(n, k), special cases of the well known split graphs, (see Definition 4.2 and [6]).

Definition 4.1. Le t n, m, p, t ∈, with 1 < t < n - 2 and 1 < p < n - t - 1. A graph with n vertices and m edges such that

is called path-complete graph, denoted PC_n,p,t , if and only if

(i) the maximal clique of PC_n,p,t is K_{n - t};

(ii) PC_n,p,1has a t-path P_t+1 = [v₀,v₁,v₂, ..., v_t such that v₀ ∈ K_{n- t}∩ P_t+1and v₁is joined to K_n - t by p edges;

(iii) there are no other edges.

In the case where t = 1, we have PC_n,p,₁ is also a split graph. The first two theorems give lower and upper bounds to the degree deviation and the variance measures of PC _n,p,₁. Moreover, it characterize extremal graphs that attained those bounds. The third theorem gives the lower and upper bounds to the imbalanced measure.

Theorem 4.1. Fo r n > 4 and ∀p, 1 < p < n - 2, we have < s(PC_n,p,₁ ) < . The lower bound is attained by PC_n,n-2,1and the upper bound is attained by PC_n,1,1.

Proof. Let PC_n,p,1 and PC_n,p+1,1 be two path complete graphs. Through a simple algebraic manipulation, we obtain

Consequently, s(PC_n,p,1) - s(PC_n,p+1,1) > . Since n > 2, s(PC_n,p,1) - s(PC_n,p+1,1) > 0, we conclude that ∀p,1 < p < n - 3, s(PC_n,p,1) > s(PC_n,p+1,1). □

Theorem 4.2. For n > 4 and 1 < p < n - 2, we have< σ(PC_n,p,1) < The lower bound is attained by PC_n,n-2,1and the upper bound by PC_n,1,1.

Proof. From the definition of the variance measure, it is easy to reach PC_n,p,1 as follows,

So, we have, σ(PC_n,p,1) - σ(PC_n,p+1,1) = > 0. Since the difference above is strictly positive, we obtain ∀p, 1 < p < n - 3, σ(PC_n,p,1) > σ(PC_n,p+1,1), σ(PC_n,1,1) = and σ(PC_n,n-2,1) =.

Theorem 4.3. Le t n > 4 and 1 < p < n - 2. The maximal v alue to irr (PC_n,p,1) is attained by p = if n is odd, and by p = or p = if n is even.

Proof. Under the conditions of n and p given by the hypothesis of the theorem, consider the graph PC_n,p,1. From the definition of the imbalanced measure, we get irr (PC_n,p,1) = 2 pn - 2 p - 2p².Let f (p) = 2 pn - 2 p - 2 p². It also immediate to prove that the maximal value of f (p) is obtained when p = ^n-1.Since p is an integer number, then n has to be odd. Otherwise, f () = f (n/2) = f () = f ((n - 2)/2) = (n² - 2n)/2 and the maximal value of f (p) is satisfied by f () and f (). □

The Figure 7 shows extremal graphs PC_n,p,1 with n = 7 and 8 vertices, where irr(PC_7,3,1) = 18 and irr(PC_8,3,1) =irr(PC_8,4,1) = 24.

Although we still have been unable to determine a simple algebraic expression for the spectral measure even for a small family of graphs such as G = PC_n,p,1, we can show some properties of the index of this graph as illustrated in Lemmas 4.1 and 4.2 which can be helpful in the future.

Lemma 4.1. Le t n > 4 and 1 < p < n -2. The spectrum of a path complete graph PC_n,p,1is given by -1 with algebraic multiplicity n - 3 and the remaining eigenvalues are the roots of the following polynomial p(x) = x³ +(3 -n)x² -(n +p -2)x -2 p +np - p².

Proof. For convenientn and p, the adjacency matrix of PC_n,p,1 is as follows:

where J denotes an all-one-matrix and I is the identity matrix. For 1 < i < n, let e_i∈ ⁿ be the i-th vector of the canonical basis. It is not too difficult to see that ∀ i, 2 < i < n -2, if v = e_i - e_n-1, then Av =- v. So, v is an eigenvector correspondent to the eigenvalue -1 with algebraic multiplicity at least n -3.

There is an orthogonal basis of n formed by the eigenvectors of A(PC_n,p,1). So, there are the eigenvectors as y^t =Then, the remaining 3 eigenvalues of A(PC_n,p,1)

are the same ones of the matrix

which are exactly the roots of P_B(x) = x³ +(3 -n)x² -nx -(p -2)x -2 p +np - p². □

Lemma 4.2. For all p, 1 <p <n -3, we have ε(PC_n,p,1) -ε(PC_{n,p +1,1}) = λ1(PCn,p,1 )- λ₁(PC_n,p+1,1) +.

Proof. It is well known that if H is a subgraph of G, their indices attained the inequality λ₁(H) < λ₁(G). The strict inequality holds if G is a connected graph and H is a proper subgraph of G. As PC_n,p,1 ⊂ PC_n,p+1,1, consequently, for 1 < p < n - 3, we have λ₁(PC_n,p,1) < λ₁(PC_n,p+1,1). Let mPC_n,p,1 and mPC_n,p+1,1 be the number of edges of those graphs, respectively. It is obvious that mPC_n,p+1,1= mPC_n,p,1 + and, as (PC_n,p,1) = , we easily have ∀ j ,1 < j < n - 3, (PC_n,j+1,1) = (PC_n,j,1)+ . So, ε(PC_n,p,1) - ε(PC_n,p+1,1) = λ₁(PC_n,p,1) - λ₁(PC_n,p+1,1) + . □

Numerically, with the aid of the software AGX, see [11], we found that the difference between the indices of the graphs PC_n,p+1,1 and PC_n,p,1 is at most . The experiments were done with graphs up to n < 25 vertices. Based on these, we establish the following conjecture.

Conjecture 4.1. For n > 4 and 1 < p < n-3, we have 0 < λ₁(PC_n,p+1,1)-λ₁₍PC_n,p,1)< .

From now on we will investigate the irregularity on the complete split graphs.

Definition 4.2.Le t k and n be integer numbers such that 0 <k <n. Agraph S(n, k) is a split graph if there is a partition of its vertex set into a clique of order k and a stable set of order n - k. A complete split graph, CS(n, k), is a split graph such that each vertex of the clique is adjacent to each vertex of the stable set.

The next two theorems characterize the most irregular complete split graphs relative to s(CS(n, k)) and irr(CS(n, k)), respectively. As one will see, both extremal graphs are the same. Besides, the theorems show the expression of the respective measures.

Theorem 4.4. Let k ∈ and 0 < k < n. If G = CS(n, k) is a complete split graph then s(G) = ² k(n - k)(n - 1 - k). Besides, among all complete split graphs, the most irregular one by s(G) has to attend the following conditions on k:

Proof. Consider G = CS(n, k). From the definition of the deviation measure, we have s(G) = k(n - k)(n - 1 - k). For a given n, define f (k) = k(n - k)(n - 1 - k). The maximal value of f (k) is obtained by k = n - - . Since k is an integer, we have to determine and . Let p, r ∈ + and 0 < r < 2 such that n = 3 p + r. So, n² - n + 1 = 9 p² + 6 pr + r² - 3 p - r + 1. Let r = 0, and rewrite n² - n + 1 = 9 p² - 3 p + 1. Thus, (3 p + 1)² - 9 p = (3 p - 1)² + 3 p and 3 p - 1 < < 3 p + 1. Therefore, we reach - 1 < k < if n mod (3) = 0. Similarly, for r = 1 and r = 2, we get < k < , if n mod (3) = 1 and, < k < , if n mod (3) = 2.

From the results above, the conditions on and follow:

Finally, after comparing f () with f () in all previous cases, we reach the following maximal values of f (k),

i) = , if n mod (3) = 0;

ii) = , if n mod (3) = 1 and

iii) = and = , if n mod (3) = 2. □

The Figure 8 shows extremal graphs CS(n, k) with n = 5 and 6 vertices, where s(CS(5, 1) = s(CS(5, 2)) = 4, 8 and CS(6, 2) = 8.

Based on the results before and through the several numerical experiments done with the aid of AGX (see [14]), we propose the conjecture below.

Conjecture 4.2.Le t (n) be the set of all connected graphs G with n vertices. So, max_G∈ (n) s(G) = s(CS(n, k)) where

Theorem 4.5. Let k ∈, and 0 < k < n. If G = CS(n, k) is a complete split graph, then irr(G) = k(n - k)(n - 1 - k). Besides, among all complete split graphs, the most irregular one relative to irr(G) has to satisfy the following conditions on k:

Proof. Since CS(n, k) has k vertices with degrees n - 1, and n - k vertices with degree k, we have irr(G) = k(n - k)(n - 1 - k). From Theorem 4.4, irr(CS(n, k)) = s(CS(n, k)), and the result holds. □

Among all complete split graphs relative to the variance measure, Theorem 4.6 provides the most irregular one.

Theorem 4.6. Let k ∈ and 0 < k < n. If G = CS(n, k) is a complete split graph, we have σ(SC(n, k)) = (k - n + 1)² (n - k) k. Besides, among all complete split graphs, the most irregular one relative to s(G) satisfies the following conditions on k:

Proof. If G = CS(n, k), from the definition of the variance measure, we get σ(SC(n, k)) = (k - n + 1)² (n - k) k. Let n be given. Define f (k) = (k - n + 1)² (n - k) k.

By elementary algebraic manipulation, we have k = n - - as the maximal value to f (k). Similarly the proof of theorem before, the result holds. □

Theorem 4.7. The eigenvalues of CS(n, k) are (k - 1) ± each one with simple multiplicity; 0 with multiplicity n - k - 1 and, -1 with multiplicity k - 1.

Proof. Let J be the all-one-matrix and I be the identity matrix. The adjacency matrix of CS(n, k) can be written as follows,

For 2 < i < k, v = e₁ -e_i, where e_i, is the i-th vector of the canonical basis, is an eigenvector of A(CS(n, k)) corresponding to the eigenvalue -1. Its multiplicity is at least k -1. Besides, note that 0 is an eigenvalue of A(SC(n, k)) to the eigenvectors x = e_j - e_n, k + 2 < j < n. In the last case, its multiplicity is at least n - k -1. The matrix A(CS(n, k)) has an eigenvector with the form y^t = . Thus, we conclude that their remaining 2 eigenvalues are

equal to the ones of

Consequently, (k -1) ± are eigenvalues of A(CS(n, k)). □

From Theorem 4.7, the spectral measure of complete split graphs with n vertices and a clique of size k is

Although Theorem 4.7 has characterized the eigenvalues to any complete split graph, three of them were implicitly determined. It follows that, the maximum of the spectral measure function is not a simple task. The numerical experiments with done (see [14]) show that, among all connected graphs with n vertices, the complete split graphs are the most irregular to the spectral measure. More specifically, they are the stars S_n = CS(n, 1), for 4 < n <11, and CS(n, 2), for 12 < n < 15.

Tables 2 and 3 summarize our results relative to extremalities concerning PC_n,p,1 and SC(n, k) graphs respectively.

5 CONCLUSION

Among all graphs with n vertices and concerning all the measures of irregularity investigated, the quasi complete graphs are amongst the most irregular ones. However, such an observation is not enough to allow us to identify which irregularity measure is more accurate. Therefore, we have decided to investigate the irregularity in more restricted classes of graphs, such as path complete graphs P_n,p,t(in t = 1 cases) and complete split graphs SC(n, k), where k is the size of the maximal clique of the graph (see Section 4). This investigation allows us to state that:

(i) as it was said before, the imbalance measure does not satisfy the definition of the function F given in Introduction. In fact, in the particular case of PC_n,p,1 graphs, the behavior of the invariant irr( PC_n,p,1) does not work as it would be expected (see the third line of Table 2);

ε(ii) the other three measures, (G), s(G) and σ(G), are not comparable. In other words, none of them is more accurate that the other.

ACKNOWLEDGMENTS

Supported by CNPq-Brazil grants 141904/2008-9, 303456/2009-4, 305516/2010-8 and 305016/2006-7.

Received June 13, 2012

Accepted March 15, 2013

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Corresponding author

Publication Dates

• Publication in this collection
  08 Nov 2013
• Date of issue
  Dec 2013

History

• Received
  13 June 2012
• Accepted
  15 Mar 2013