International Journal of Combinatorics

International Journal of Combinatorics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 148690 | 16 pages | https://doi.org/10.1155/2014/148690

Integral Eigen-Pair Balanced Classes of Graphs with Their Ratio, Asymptote, Area, and Involution-Complementary Aspects

Academic Editor: Cai Heng Li
Received26 May 2014
Accepted03 Sep 2014
Published23 Sep 2014

Abstract

The association of integers, conjugate pairs, and robustness with the eigenvalues of graphs provides the motivation for the following definitions. A class of graphs, with the property that, for each graph (member) of the class, there exists a pair of nonzero, distinct eigenvalues, whose sum and product are integral, is said to be eigen-bibalanced. If the ratio is a function , of the order of the graphs in this class, then we investigate its asymptotic properties. Attaching the average degree to the Riemann integral of this ratio allowed for the evaluation of eigen-balanced areas of classes of graphs. Complete graphs on vertices are eigen-bibalanced with the eigen-balanced ratio which is asymptotic to the constant value of −1. Its eigen-balanced area is —we show that this is the maximum area for most known classes of eigen-bibalanced graphs. We also investigate the class of eigen-bibalanced graphs, whose class of complements gives rise to an eigen-balanced asymptote that is an involution and the effect of the asymptotic ratio on the energy of the graph theoretical representation of molecules.

1. Introduction: Integers, Conjugate Pairs, and Eigenvalues of a Graph

The graphs in this paper are simple and connected and all graph properties mentioned will be according to definitions in Harris et al. [1]. There has been much work done in the analysis of eigenvalues of matrices which are adjacency matrices of associated graphs. The following are some examples of these findings, with the references as specified.(i)There has been interest in classes of graphs whose pairs of eigenvalues satisfy certain conditions. In Sarkar and Mukherjee [2], graphs are considered with reciprocal pairs of eigenvalues , whose product is the integer 1.(ii)Pairs of eigenvalues (1, −1), summing to 0, and whose product is −1, are considered in Dias [3].(iii)Summing the eigenvalues of the adjacency matrix of a graph is connected to the energy of physical structures; see Aimei et al. [4].(iv)In the paper by van Dam [5], on regular graphs with 4 eigenvalues, he considers the eigenvalue pair of real conjugates and shows that if a matrix has an eigenvalue , then it has an eigenvalue of the same multiplicity, and vice versa. Adding the pair of conjugates and , we obtain the integer . Their product is which is integral, provided the numerator is a multiple of 4. The paper shows that there are graphs whose matrices have conjugate pairs of eigenvalues whose sum does not necessarily sum to the same integer .

The following references show other areas of research which, together with the results on eigenvalues, provide motivation for the new definitions which are contained in this paper.(i)There has been interest in the importance of pairs of numbers, whose sum and product produce the same integral constant, and this exists outside the linear algebra of matrices; see, for example, Dias [3].(ii)In Brouwer and Haemers [6], integral trees (where the eigenvalues of trees are integral) are investigated.(iii)In the cryptography article, Hamada [7] considers the conjugate code pair consisting of linear codes and satisfying the constant (integral) sum term , where is the dimension of the vector space involved and is the -digit secret information sent.(iv)In the paper by Kadin [8] he investigates the Cooper pair of opposite wave vectors and which balance by summing to 0 and whose product is .(v)Hinch and Leal [9] consider the notion of an isolated particle in the absence of rotary Brownian motion, under the condition that the hydrodynamic and external field couples exactly balance one another.(vi)Armstrong [10] investigates the importance of the quadratic part of a characteristic equation which has the form . This quadratic gives rise to the two eigenvalues . The sum and product (,, resp.) are often referred to as the eigen-pair, but we will focus on the pair of eigenvalues as the eigen-pair.

Generally, there often exist two eigenvalues (associated with the adjacency matrix of a graph) whose sum or product is integral It is therefore possible to get the same integer when adding or multiplying two distinct, nonzero eigenvalues. This integer is either a fixed constant or a function of an inherent property of the graph, for all graphs belonging to a certain class of graphs. For example, complete graphs on vertices have a pair of eigenvalues with sum of and product of for each , and the complete bipartite graphs on vertices have eigen-pair sum (of nonzero eigenvalues) of 0 and product of .

Definition 1 (function of a member of a class of graphs). We define a function of a member belonging to a class of graphs as a real function of an inherent property of the member in the class, such as the number of vertices or the clique number of a graph, and so forth.
In this paper, we combine the ideas of a pair of eigenvalues and their balanced integral sum and product with respect to a class of graphs, to introduce a definition which is a form of integral-eigenvalue balance associated with classes of graphs. We investigate classes of graphs on vertices with pairs of distinct nonzero eigenvalues such that or , where are the same integer (resp.) for each graph in the class or the same function for each graph in the class.

2. Integral Sum Eigen-Pair Balanced Classes of Graphs

Definition 2 (()-pair (integral) balanced). The class of connected graphs on elements is said to be ()-pair (integral) balanced if there exists a pair of distinct nonzero eigenvalues of the matrices associated with each class of the structures such that is the same integer as a fixed constant for each member in the class or is the same integer as a function of each member in the class. The sum balance is exact if is the same integer as a fixed constant for each member in the class, or otherwise it is nonexact.

The following are some examples of such classes of graphs, noting that sum in the examples below.

2.1. Complete Graphs

The distinct eigenvalues of the complete graph are and (as per Brouwer and Haemers [6]), with the sum of the eigenvalues being . Therefore the class of complete graphs is nonexact sumeigen-pair balanced, for .

2.2. Complete Bipartite Graphs

The class of complete bipartite graphs on vertices has as its associated eigenvalues , , and (as per Brouwer and Haemers [6]), so it is exact sumeigen-pair balanced.

The class of complete bipartite graphs on vertices, , has eigenvalues , , (as per Brouwer and Haemers [6]), so it is exact sumeigen-pair balanced (this includes the star graphs with radius 1).

2.3. Cycle Graphs

From Brouwer and Haemers [6], the cycle on vertices and edges has eigenvalues

The 3-cycle (complete graph on 3 vertices) has distinct eigenvalues −1 and 2.

The 4-cycle has distinct eigenvalues 2, 0, and −2.

The 5-cycle has distinct eigenvalues , , and  .

The 6-cycle has distinct eigenvalues , , , and .

The 7-cycle has distinct eigenvalues 2, 1.247, −0.445, and 1.802.

For the 3-, 5-, and 6-cycle, there exist two distinct nonzero eigenvalues whose sum is 1. However, for the 7-cycle, there are no two distinct eigenvalues whose sum is 1. Therefore the class of cycles is not sum(1)eigen-pair balanced as it does not satisfy Definition 2.

However, even cycles are sumeigen-pair balanced, since if , Then and yield eigenvalues of 2 and −2, respectively.

2.4. Path Graphs

From Brouwer and Haemers [6], the path on vertices and edges has eigenvalues Note that so that the nonzero pair, with and , has the sum so the class of path graphs is exact sumeigen-pair balanced.

2.5. Graph Which Is the Join of Two Graphs Whose Adjacency Matrices Are Both Circulant Matrices

From Lee and Yeh [11], the conjugate eigenvalues of the join of two circulant matrices of graphs are So the sum of the eigenvalues is so the class of graphs, which are the join of two graphs whose adjacency matrices are circulant, is eigen-pair balanced.

2.6. Wheel Graphs

As per Lee and Yeh [11], the wheel graph on vertices, with spokes (join of cycle with a single vertex), has conjugate eigenvalues The sum of the conjugate eigenvalues is therefore so the class of cycle wheel graphs is exact sumeigen-pair balanced.

2.7. Strongly Regular Graphs

If a connected regular graph of degree with parameters is strongly regular (i.e., ), then has at least 3 different eigenvalues. The eigenvalues are See Spielman [12].

The complement of an is also strongly regular. It is an , .

Note that if we ignore the largest eigenvalue of strongly regular graphs, adding the remaining two eigenvalues yields the integer so the class of strongly regular graphs with parameters is nonexact sumeigen-pair balanced; see Godsil and Royle [13] for more information on strongly regular graphs.

2.8. Divisible Design Graphs

Definition 3 (divisible design graph). A -regular graph is a divisible design graph if the vertex set can be partitioned into classes of size , such that two distinct vertices from the same class have exactly common neighbours, and two vertices from different classes have exactly common neighbours.
The eigenvalues of divisible design graphs are provided in Haemers [14]; there are 5 distinct eigenvalues. Two of the eigenvalues are so the sum of the eigen-pair is Therefore, the class of divisible design graphs is exact sumeigen-pair balanced.

2.9. Hypercube Graphs

As per Brouwer and Haemears [6], the -regular hypercube on vertices and edges has eigenvalues Using the eigenvalues and , for and , this class of graphs will be sumeigen-pair balanced.

2.10. Eigenvalue Pair of Real Conjugates

The following results are due to van Dam and Haemers [15].

By and we denote the rings of polynomials over the integer and rational numbers, respectively.

Lemma 4. If a monic polynomial has a monic divisor , then also .

Lemma 5. If , with , is an irrational root of a polynomial , then so is , with the same multiplicity.

The characteristic polynomial of the adjacency matrix of a graph is monic and has integral coefficients. Using Lemmas 4 and 5 we now obtain the following results.

Corollary 6. Every rational eigenvalue of a graph is integral.

Corollary 7. If is an irrational eigenvalue of a graph, for some , then so is , with the same multiplicity, and .

Adding the pair of real conjugates , we obtain the integer .

Therefore, if the real conjugate pairs are eigenvalues associated with the adjacency matrix of all graphs belonging to a class of graphs, then the class of graphs is eigen-pair balanced.

3. Integral Product Eigen-Pair Balanced Classes of Graphs

Definition 8 (()-pair (integral) balanced). A class of connected graphs on elements is said to be ()-pair (integral) balanced if there exists a pair of of distinct nonzero eigenvalues (counting eigenvalues only once, i.e., ignoring multiplicities) of the matrices associated with each class of the structures such that is the same integer as a fixed constant for each member in the class or is the same integer as a function of each member in the class. The product balance is exact if is the same integer as a fixed constant for each member in the class; otherwise it is nonexact.

The following are some examples of such classes of product balanced classes of graphs, discussed above, noting that the product of the pair in the examples below.

3.1. Complete Graphs

The class of complete graphs is nonexact producteigen-pair balanced for , since the eigenvalues of the associated adjacency matrices, are, as per Section 2.1, and .

3.2. Complete Bipartite Graphs

The class of complete bipartite graphs on vertices has as its associated eigenvalues , , and 0 (as per Section 2.2), so that they are nonexact producteigen-pair balanced.

The class of complete bipartite graphs on vertices, , has distinct eigenvalues , , and (as per Section 2.2), so that it is nonexact producteigen-pair balanced (this includes the star graphs with radius 1).

3.3. Cycle Graphs

Cycles graphs on vertices have associated eigenvalues

on 3 vertices has eigenvalues 2 and −1 so that the eigen-pair product is −2.

on 4 vertices has eigenvalues 2, 0, and −2 so that the eigen-pair product is −4.

on 5 vertices has eigenvalues 2, , and with conjugate eigen-pair product −1.

on 6 vertices has eigenvalues 2, 1, −1, and −2. Possible eigen-pair products are −1, −2, 2, and −4.

The 7-cycle has eigenvalues 2, 1.247, −0.445, and −1.802. No product of two eigenvalues yields an integer!

Therefore, the class of cycles is not eigenproduct balanced for any integer .

However, the class of even cycles is producteigen-pair balanced, since if then so that for we get eigenvalue 2 and for we get eigenvalue −2, with eigen-pair product −4.

3.4. Path Graphs

Paths graphs on vertices have eigenvalues: From Section 2.4, so that with and the product is which is a function of but is not integral in general.

3.5. Graph Which Is the Join of Two Graphs Whose Adjacency Matrices Are Both Circulant Matrices

The conjugate eigenvalues are So the product of the eigenvalues is so the class of graphs whose adjacency matrix is the join of two graphs whose adjacency matrices are circulant matrices is producteigen-pair balanced.

3.6. Wheel Graphs

The wheel graph on vertices, with spokes, has conjugate eigenvalues The product of the conjugate eigenvalues is so the class of cycle wheel graphs is nonexact producteigen-pair balanced.

3.7. Strongly Regular Graphs

The conjugate eigen-pair of strongly regular graphs is If we multiply the two conjugate pairs of strongly regular graphs we obtain the integer , so that the class of strongly regular graphs is nonexact producteigen-pair balanced.

3.8. Divisible Design Graphs

Two of the eigenvalues of a divisible design graph are This class of graphs has eigen-pair product Therefore, the class of divisible design graphs is nonexact productbalanced.

3.9. Bipartite Graphs with Four Distinct Eigenvalues

The eigenvalues of a bipartite graph with four distinct eigenvalues are Then product and product .

Therefore, incidence graphs of symmetric 2- designs are eigen-pair balanced for and of the nonexact kind.

3.10. Hypercube Graphs

The class of -regular hypercubes on vertices and edges has eigenvalues: Using the eigenvalues and , for , , this class of graphs is producteigen-pair balanced.

3.11. Eigenvalue Pair of Real Conjugates

The product of the real conjugate pair of eigenvalues and is . This is integral, provided the numerator is a multiple of 4.

Therefore, if the real conjugate pairs are eigenvalues associated with the adjacency matrix of all graphs belonging to a class of graphs, then the class of graphs is producteigen-pair balanced.

4. Eigen-Bibalanced Classes of Graphs

Definition 9 (eigen-bibalanced classes of graphs). Classes of connected graphs, which are both sum and product eigen-pair balanced, are said to be eigen-bibalanced with respect to the eigen-pair . If this pair is unique to the class, then it is uniquely eigen-bibalanced. For example, the class of complete graphs is uniquely eigen-bibalanced with respect to the eigen-pair .

The largest eigenvalue occurs in the eigen-pair associated with some classes of graphs discussed above. We observe the following.(i)The only regular eigen-pair balanced graphs on 2 and 3 vertices are and .(ii)The 4-cycle is the same as the complete bipartite graph , which is sum and product eigen-pair balanced.(iii)The only other regular graph on 4 vertices is .(iv)The 5-cycle has eigenvalues , , and which are not sum or product eigen-pair balanced when the largest eigenvalue is included in the eigen-pair.(v)The only other regular graph on 5 vertices is .

Thus we have the following theorem.

Theorem 10. The only regular graphs on vertices, where , belonging to eigen-pair balanced classes of graphs, where the eigen-pair contains the largest eigenvalue, are , , , , and .

5. Eigen-Bibalanced Classes of Graphs: Criticality, Ratios, Asymptotes, and Area

If a class of connected graphs are both sum and product eigen-pair balanced with respect to the eigen-pair , they have been defined above as eigen-bibalanced with respect to . The class of complete graphs is eigen-bibalanced with the property that the removal of any vertex from results in a complete graph, which belongs to the same class of complete graphs, which is eigen-bibalanced. The same holds for complete bipartite graphs except for star graphs. Such graphs are said to be stable eigen-bibalanced.

Definition 11 (critically eigen-bibalanced classes of graphs). If belongs to a class of eigen-bibalanced graphs, and there exists a vertex of , such that belongs to a class of graphs which is not eigen-bibalanced, we say that is critically eigen-bibalanced with respect to .
Wheels on spokes are eigen-bibalanced and the removal of the central vertex results in -cycles, which are not eigen-bibalanced. Therefore, the class of wheel graphs is critically eigen-bibalanced with respect to its central vertex. This suggests that the central vertex is essential to the eigen-bibalanced characteristic of wheels.
The reciprocals of eigenvalues are connected to the idea of robustness or tightness of graphs, Brouwer and Haemers [7]. Since and are nonzero, the sum of their reciprocals is defined. Therefore we have the following definition.

Definition 12 (eigen-bibalanced ratio of classes of graphs). The importance of ratios in graphs is well researched (see Winter and Adewusi [16]). The eigen-bibalanced ratio of the class of graphs (with respect to the eigen-pairs ) is As and are nonzero, the product is never zero, and so this ratio will always be defined.

Definition 13 (eigen-bibalanced ratio asymptote of classes of graphs). If this ratio is a function of the order of the graph and has a horizontal asymptote (see Winter and Adewusi [16]), we call this asymptote the eigen-bibalanced ratio asymptote with respect to the eigen-pair and is denoted by This asymptote can be seen as describing the behavior of the ratio as the order of the graph becomes increasingly large.
The “area” term can be found in the following relation involving spanning trees. Let be the matrix composed of the degrees of in the main diagonal form by adding 1 to each entry of . We then form the shadow number of a graph defined by , where is the adjacency matrix of . We then have the combinatorial result where is the number of spanning trees of a connected graph .
Also, the number of spanning trees of a connected graph is associated with the Laplacian eigenvalues, , of the graph by the following: This excludes the first Laplacian eigenvalue. Thus, as in the case of the complete graph, the eigenvalue of the adjacency matrix associated with will not be taken into account when considering spanning trees.
Eigenvalues have been associated with the expansion of graphs (see Hamada [7]), which motivates the idea of areas associated with a class of graphs.
If the eigen-bibalanced ratio of a class of graphs is a function of , then we are able to integrate it with respect to , which leads to the following definition.

Definition 14 (eigen-bibalanced ratio area of classes of graphs). We define the eigen-bibalanced ratio area of the class of graphs with respect to the eigen-pair as (see Winter and Adewusi [16]) where is the number of edges and is the number of vertices, and when , 1, or 2.
Now we define length denoted by , as
, that is, the average degree of the vertices in , and define height, denoted by , as so .
If there is more than one pair giving rise to such area, then the area of the class is for all pairs . If there is only one eigen-pair associated with the class of graphs that gives rise to the area, then the area is unique.
The height involves binding the sum of the reciprocals of the eigen-pair by its integration and the multiplication of this height by the average degree. This involves one of the most basic, yet important, combinatorial aspects of the graph and results in the term appearing in the eigen-bibalanced ratio area of some classes of graphs.

6. Examples of Eigen-Bibalanced Classes of Graphs

When we refer to a graph having eigen-pair balanced properties such as sum, product, bibalanced, ratio, and asymptote, we imply that belongs to a class of graphs having such eigen-pair properties. We will now look at various examples of eigen-bibalanced classes of graphs.

6.1. Complete Graphs

The complete graph on vertices has the unique eigen-bibalanced ratio of This depends on the order of the graph and has the unique eigen-bibalanced ratio asymptote and eigen-bibalanced ratio area where When we have so so that its area is Note that the length of the longest path for the complete graph is , so that in the above expression can be regarded as the height of the graph. Also, the term occurs as part of the upper bound of the diameter of a graph involving the second largest eigenvalue; see Brouwer and Haemers [6].

Is this area the maximum for all classes of eigen-bibalanced graphs?

6.2. Complete Bipartite Graphs

The complete bipartite graph on vertices has the eigen-bibalanced ratio of which is independent of the size of the graph.

Its area is This attains its maximum when , and then the graph (the complete split bipartite graph on vertices) is -regular and the area is

6.3. Wheel Graphs

Wheels on vertices, containing spokes and edges, have eigen-bibalanced ratio This depends on the size of the graph, so they have an eigen-bibalanced ratio asymptote of 0 and eigen-bibalanced ratio area of Now when so that .

So

6.4. Star Graphs

As per Brouwer and Haemers [6], star graphs with rays of length 2 have eigenvalues , , , and .

Using the pair we obtain the ratio and using the pair we get Using the class of graphs, where and eigen-pair , we have the ratio Therefore the class of star graphs with rays of length 2 does not have a unique eigen-bibalanced ratio.

The area with respect to the pair is and with respect to the pair is Since , the areas are, respectively, The greater of the two gives the area of the class of graphs; that is,

6.5. Hypercube Graphs

The -regular hypercube has eigenvalues , , and eigen-bibalanced ratio ( fixed, varying, , and ) where, since , we have .

Then For and using the eigenvalue pair and , the eigen-bibalanced ratio area is Setting and rearranging, we get So Also, for , we have Now substituting the above results into (55), we get .

Now and so that Although this is not a good approximation for the area of a hypercube class of graphs, it may suggest that the area of such a class of graphs is greater than that of complete graphs.

6.6. Join of Two Graphs

Taking the join of the complement of the complete graph on 2 vertices and the complete graph on vertices, we get that the resulting adjacency matrix of the graph has the conjugate pair of eigenvalues so that their eigen-bibalanced ratio is

6.7. Cycle Graphs

Conjecture 15. The only class of regular graphs which are neither sum nor product eigen-pair balanced is cycles.

7. Eigen-Bibalanced Properties of the Class of Complements of Graphs

Theorem 16. Let be a class of eigen-bibalanced, -regular graphs with eigen-pair ; or . Let be the class of graphs consisting of the complement, , of graphs , where is connected. Then, for all , is -regular, eigen-bibalanced with eigen-pair , and the eigen-bibalanced ratio of is . Therefore, the class of graphs is eigen-bibalanced.

Proof. As per Brouwer and Haemers [6], if are eigenvalues of and or −1, then and are eigenvalues of .
Therefore, Since is eigen-bibalanced, then and are constant integers. Therefore, is eigen-bibalanced.
Therefore, Therefore the class of graphs is eigen-bibalanced.

Corollary 17. Let be a class of eigen-bibalanced, -regular graphs with eigen-pair ; or −1. Let be the class of graphs consisting of the complement, , of graphs , where is connected. If the asymptote of the ratio , or and , then

Proof. Consider the following: Now

Corollary 18. Let be a class of eigen-bibalanced -regular graphs with eigen-bibalanced ratio asymptote with respect to pair where or −1. Let be the class of graphs consisting of the complement, , of graphs , where is connected. Then the eigen-bibalanced ratio asymptote of with respect to eigen-pair is which is an involution.

Proof. As per Corollary 17, let .
Then, so that which implies that .
Functions, which are equal to their own inverse, are called involutions, so that is an involution. Classes of uniquely eigen-bibalanced graphs whose complements form a class of eigen-bibalanced graphs with an involution property are said to be involution-complementary.

Corollary 19. The involution is a solution of the differential equation

Proof. Since then Also, , Differentiating both sides, we get Since is an involution, , then differentiating both sides and recalling we get Substituting (71) and (74) into (73), we get where .
Solving this variable separable equation we get and is a general solution of (69).
If and , then the solution is if , then , so

In Section 6.6 we showed that is the eigen-bibalanced ratio asymptote of the class of graphs comprising the join of the complement of the complete graph on 2 vertices and the complete graph on vertices.

Note. There are other possible solutions of the general differential equation , .

For example, when , is a solution of the differential equation

8. Properties of Eigen-Bibalanced Classes of Graphs

Theorem 20. If a class of noncomplete graphs is eigen-bibalanced with respect to the eigen-pair , are conjugate eigen-pairs arising from the quadratic , with at least one of positive and of the form (with being an integer and being negative), and the ratio is a function of , then is negative and the eigen-pair balanced ratio asymptote lies on the interval .

Proof. Let the conjugate eigen-pair arise from the roots of the quadratic ; that is, Assume , then : and , and , which is a contradiction of the assumption that at least one of is positive.
Therefore, we have shown that .
If , then and either or , which is not allowed.
Therefore, we have shown that .
So let , where .
We have and (as these are conjugate eigenvalues of the class of noncomplete graphs ); ; and and If then the ratio . However, we are given that is a function of , so we must have .
If and are both fixed constants, then the ratio is not a function of , so we cannot have and being both fixed constants.
From (82) above, . If , is of order , and is a fixed constant , then If is a function of , so will be a function of .
If both and are functions of , then is and has where .
Therefore, .
Now let us assume that and ,   negative (as per conditions in Theorem 20).
Then if , , , and and are constants, then Since is an integer, must be an integer too (as they are conjugate pairs), so . Therefore, So we have proved that .
We have therefore shown separately that and under different conditions.
Therefore .

For the complete graph , the quadratic for the complete graph, with eigenvalues , , and product , is So, .

So the eigen-bibalanced ratio asymptote of the complete graphs is −1, which is the same as one of the eigen-pairs. If a class of graphs is eigen-bibalanced with respect to the pair and its asymptote is the same as one of the eigen-pairs, then it is said to be asymptotically closed with respect to the pair . Therefore, the class of complete graphs is asymptotically closed with respect to the pair .

Theorem 21. The eigen-bibalanced ratio areas of complete bipartite graphs, wheel graphs, and star graphs with rays of length 2 are each bounded above by the area of the complete graph.

Proof. As discussed above, the eigen-bibalanced ratio area of the complete graph is As per Section 6.2, the eigen-bibalanced ratio area of the split complete bipartite graph is As per Section 6.3, the eigen-bibalanced ratio area of the wheel graph is As per Section 6.4, the eigen-bibalanced ratio area of the star graph with rays of length 2 is where and .
(1) Considering the eigen-bibalanced ratio area of the complete graph and the complete split bipartite graph, we get Table 1.
Now, for large values of , behaves like and is an increasing function of , and