New families of mean graphs.
Abstract:
Let G(V, E) be a graph with p vertices and q edges. A vertex labeling of G is an assignment f : V(G) [right arrow] {1, 2, 3, ... ,p + q} be an injection. For a vertex labeling f, the induced Smarandachely edge m-labeling [f.sup.*.sub.s] for an edge e = uv, an integer m [greater than or equal to] 2 is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then f is called a Smarandachely super m-mean labeling if f (V(G)) [union] {[f.sup.*] (e) : e [member of] E(G)} = {1, 2, 3, ... ,p + q}. Particularly, in the case of m = 2, we know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Such a labeling is usually called a super mean labeling. A graph that admits a Smarandachely super mean m-labeling is called Smarandachely super m-mean graph, particularly, super mean graph if m = 2. In this paper, we discuss two kinds of constructing larger mean graphs. Here we prove that ([P.sub.m]; [C.sub.n])m [greater than or equal to] 1, n [greater than or equal to] 3, ([P.sub.m]; [Q.sub.3])m [greater than or equal to] 1,([P.sub.2n]; [S.sub.m])m [greater than or equal to] 3, n [greater than or equal to] 1 and for any n [greater than or equal to] 1 ([P.sub.n]; [S.sub.1]), ([P.sub.n]; [S.sub.2]) are mean graphs. Also we establish that [[P.sub.m]; [C.sub.n]]m [greater than or equal to] 1, n [greater than or equal to] 3, [[P.sub.m]; [Q.sub.3]]m [greater than or equal to] 1 and [[P.sub.m]; [C.sup.(2).sub.n]]m [greater than or equal to] 1, n [greater than or equal to] 3 are mean graphs.

Key Words: Labeling, mean labeling, mean graphs, Smarandachely edge m-labeling, Smarandachely super m-mean labeling, super mean graph.

AMS(2000): 05C78

Article Type:
Report
Subject:
Graph theory (Research)
Number theory (Research)
Average (Research)
Authors:
Vasuki, R.
Pub Date:
07/01/2010
Publication:
Name: International Journal of Mathematical Combinatorics Publisher: American Research Press Audience: Academic Format: Magazine/Journal Subject: Mathematics Copyright: COPYRIGHT 2010 American Research Press ISSN: 1937-1055
Issue:
Date: July, 2010 Source Volume: 2
Topic:
Event Code: 310 Science & research
Geographic:
Geographic Scope: India Geographic Code: 9INDI India
Accession Number:
234934920
Full Text:
[section] 1 . Introduction

Throughout this paper, by a graph we mean a finite, undirected, simple graph. Let G(V, E) be a graph with p vertices and q edges. A path on n vertices is denoted by [P.sub.n] and a cycle on n vertices is denoted by [C.sub.n]. The graph [P.sub.2] x [P.sub.2] x [P.sub.2] is called the cube and is denoted by [Q.sub.3]. For notations and terminology we follow [1].

A vertex labeling of G is an assignment f : V(G) [right arrow] {1, 2, 3, ... ,p + q} be an injection. For a vertex labeling f, the induced Smarandachely edge m-labeling [f.sup.*.sub.S] for an edge e = uv, an integer m [greater than or equal to] 2 is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then f is called a Smarandachely super m-mean labeling if f (V(G)) [union] {[f.sup.*](e) : e [member of] E(G)} = {1, 2, 3, ... ,p + q} . Particularly, in the case of m = 2, we know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Such a labeling is usually called a super mean labeling. A graph that admits a Smarandachely super mean m-labeling is called Smarandachely super m-mean graph, particularly, super mean graph if m = 2. The mean labeling of the Petersen graph is given in Figure 1.

[FIGURE 1 OMITTED]

A super mean labeling of the graph [K.sub.2,4] is shown in Figure 2.

[FIGURE 2 OMITTED]

The concept of mean labeling was first introduced by Somasundaram and Ponraj [2] in the year 2003. They have studied in [2-5,8-9], the meanness of many standard graphs like [P.sub.n], [C.sub.n], [K.sub.n](n [less than or equal to] 3), the ladder, the triangular snake, [K.sub.12], [K.sub.13], [K.sub.2,n], [K.sub.2] + m[K.sub.1], [K.sup.c.sub.n] + 2[K.sub.2], [S.sub.m] + [K.sub.1], [C.sub.m] [union] [P.sub.n](m [greater than or equal to] 3, n [greater than or equal to] 2), quadrilateral snake, comb, bistars B(n), [B.sub.n+1n], [B.sub.n+2,n], the carona of ladder, subdivision of central edge of [B.sub.n,n], subdivision of the star [K.sub.1,n](n [less than or equal to] 4), the friendship graph [C.sup.(2).sub.3], crown [C.sub.n] [dot encircle] [K.sub.1], [C.sup.(2).sub.n], the dragon, arbitrary super subdivision of a path etc. In addition, they have proved that the graphs [K.sub.n](n > 3), [K.sub.1,n](n > 3), [B.sub.m,n](m > n + 2), S([K.sub.1,n])n > 4, [C.sup.(t).sub.3](t > 2), the wheel [W.sub.n] are not mean graphs.

The concept of super mean labeling was first introduced by R. Ponraj and D. Ramya [6]. They have studied in [6-7] the super mean labeling of some standard graphs. Also they determined all super mean graph of order [less than or equal to] 5. In [10], the super meanness of the graph [C.sub.2n] for n [greater than or equal to] 3, the H-graph, Corona of a H-graph, 2-corona of a H-graph, corona of cycle [C.sub.n] for n [greater than or equal to] 3, m[C.sub.n]-snake for m [greater than or equal to] 1, n [greater than or equal to] 3 and n [not equal to] 4, the dragon [P.sub.n]([C.sub.m]) for m [greater than or equal to] 3 and m [not equal to] 4 and [C.sub.m] x [P.sub.n] for m = 3, 5 are proved.

Let [C.sub.n] be a cycle with fixed vertex v and ([P.sub.m]; [C.sub.n]) the graph obtained from m copies of [C.sub.n] and the path [P.sub.m]: [u.sub.1][u.sub.2] ... [u.sub.m] by joining [u.sub.i] with the vertex v of the ith copy of [C.sub.n] by means of an edge, for 1 [less than or equal to] i [less than or equal to] m.

Let [Q.sub.3] be a cube with fixed vertex v and ([P.sub.m]; [Q.sub.3]) the graph obtained from m copies of [Q.sub.3] and the path [P.sub.m]: [u.sub.1][u.sub.2] ... [u.sub.m] by joining [u.sub.i] with the vertex v of the ith copy of [Q.sub.3] by means of an edge, for 1 [less than or equal to] i [less than or equal to] m.

Let [S.sub.m] be a star with vertices [v.sub.0], [v.sub.1], [v.sub.2], ... ,[v.sub.m]. Define ([P.sub.2n]; [S.sub.m]) to be the graph obtained from 2n copies of [S.sub.m] and the path [P.sub.2n]: [u.sub.1][u.sub.2] ... [u.sub.2n] by joining [u.sub.j] with the vertex [v.sub.0] of the jth copy of [S.sub.m] by means of an edge, for 1 [less than or equal to] j [less than or equal to] 2n, ([P.sub.n]; [S.sub.1]) the graph obtained from n copies of [S.sub.1] and the path [P.sub.n]: [u.sub.1][u.sub.2] ... [u.sub.n] by joining [u.sub.j] with the vertex [v.sub.0] of the jth copy of [S.sub.1] by means of an edge, for 1 [less than or equal to] j [less than or equal to] n, ([P.sub.n]; [S.sub.2]) the graph obtained from n copies of [S.sub.2] and the path [P.sub.n]: [u.sub.1][u.sub.2] ... [u.sub.n] by joining [u.sub.j] with the vertex [v.sub.0] of the jth copy of [S.sub.2] by means of an edge, for 1 [less than or equal to] j [less than or equal to] n.

Suppose [C.sub.n]: [v.sub.1][v.sub.2] ... [v.sub.n][v.sub.1] be a cycle of length n. Let [[P.sub.m]; [C.sub.n]] be the graph obtained from m copies of [C.sub.n] with vertices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and joining [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by means of an edge, for some j and 1 [less than or equal to] i [less than or equal to] m - 1.

Let [Q.sub.3] be a cube and [[P.sub.m]; [Q.sub.3]] the graph obtained from m copies of [Q.sub.3] with vertices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the path [P.sub.m]: [u.sub.1][u.sub.2] ... [u.sub.m] by adding the edges [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [C.sup.(2).sub.n] be a friendship graph. Define [[P.sub.m]; [C.sup.(2).sub.n]] to be the graph obtained from m copies of [C.sup.(2).sub.n] and the path [P.sub.m]: [u.sub.1][u.sub.2] ... [u.sub.m] by joining [u.sub.i] with the center vertex of the ith copy of [C.sup.(2).sub.n] for 1 [less than or equal to] i [less than or equal to] m.

In this paper, we prove that ([P.sub.m]; [C.sub.n])m [greater than or equal to] 1, n [greater than or equal to] 3, ([P.sub.m]; [Q.sub.3])m [greater than or equal to] 1, ([P.sub.2n]; [S.sub.m])m [greater than or equal to] 3, n [greater than or equal to] 1, and for any n [greater than or equal to] 1([P.sub.n]; [S.sub.1]), ([P.sub.n]; [S.sub.2]) are mean graphs. Also we establish that [[P.sub.m]; [C.sub.n]]m [greater than or equal to] 1, n [greater than or equal to] 3, [[P.sub.m]; [Q.sub.3]]m [greater than or equal to] 1 and [[P.sub.m]; [C.sup.(2).sub.n]]m [greater than or equal to] 1, n [greater than or equal to] 3 are mean graphs.

[section]2. Mean Graphs ([P.sub.m]; G)

Let G be a graph with fixed vertex v and let ([P.sub.m]; G) be the graph obtained from m copies of G and the path [P.sub.m]: [u.sub.1][u.sub.2] ... [u.sub.m] by joining [u.sub.i] with the vertex v of the ith copy of G by means of an edge, for 1 [less than or equal to] i [less than or equal to] m.

For example ([P.sub.4]; [C.sub.4]) is shown in Figure 3.

[FIGURE 3 OMITTED]

Theorem 2.1 ([P.sub.m]; [C.sub.n]) is a mean graph, n [greater than or equal to] 3.

Proof Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the vertices in the ith copy of [C.sub.n], 1 [less than or equal to] i [less than or equal to] m and [u.sub.1], [u.sub.2], ... ,[u.sub.m] be the vertices of [P.sub.m]. Then define f on V([P.sub.m]; [C.sub.n]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Label the vertices of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as follows:

Case (i) n is odd

When i is odd,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When i is even,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Case (ii) n is even

When i is odd,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When i is even,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edge ujuj+1 is (n + 2)i, 1 [less than or equal to] i [less than or equal to] m - 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the label of the edges of the cycle are

(n + 2)i - 1, (n + 2)i - 2, ... ,(n + 2)i - n if i is odd,

(n + 2)i - 2, (n + 2)i - 3, ... ,(n + 2)i - (n + 1) if i is even.

For example, the mean labelings of ([P.sub.6]; [C.sub.5]) and ([P.sub.5]; [C.sub.6]) are shown in Figure 4.

[FIGURE 4 OMITTED]

Theorem 2.2 ([P.sub.m]; [Q.sub.3]) is a mean graph.

Proof For 1 [less than or equal to] j [less than or equal to] 8, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the vertices in the ith copy of [Q.sub.3], 1 [less than or equal to] i [less than or equal to] m and [u.sub.1], [u.sub.2], ... ,[u.sub.m] be the vertices of [P.sub.m].

Then define f on V([P.sub.m]; [Q.sub.3]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When i is odd,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when i is even,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edges of [P.sub.m] are 14i, 1 [less than or equal to] i [less than or equal to] m - 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edges of the cube are

14i - 1,14i - 2, ... ,14i - 12 if i is odd,

14i - 2,14i - 3, ... ,14i - 13 if i is even.

For example, the mean labeling of ([P.sub.5]; [Q.sub.3]) is shown in Figure 5.

[FIGURE 5 OMITTED]

Theorem 2.3 ([P.sub.2n]; [S.sub.m]) is a mean graph, m [greater than or equal to] 3, n [greater than or equal to] 1.

Proof Let [u.sub.1], [u.sub.2], ... ,[u.sub.2n] be the vertices of [P.sub.2n]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the vertices in the jth copy of [S.sub.m], 1 [less than or equal to] j [less than or equal to] 2n.

Label the vertices of ([P.sub.2n]; [S.sub.m]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edge [u.sub.j][u.sub.j+1] is (m + 2)j, 1 [less than or equal to] j [less than or equal to] 2n - 1

The label of the edge [u.sub.j][vo.sub.j] is

The label of he edge [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For example, the mean labeling of ([P.sub.6]; [S.sub.5]) is shown in Figure 6. ?

[FIGURE 6 OMITTED]

Theorem 2.4 ([P.sub.n]; [S.sub.1]) and ([P.sub.n]; [S.sub.2]) are mean graphs for any n [greater than or equal to] 1.

Proof Let [u.sub.1], [u.sub.2], ... ,[u.sub.n] be the vertices of [P.sub.n]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the vertices of [S.sub.1].

Label the vertices of ([P.sub.n]; [S.sub.1]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edges of [P.sub.n] are 3j, 1 [less than or equal to] j [less than or equal to] n - 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the vertices of [S.sub.2].

Label the vertices of ([P.sub.n]; [S.sub.2]) as follows:

The label of the edges of [P.sub.n] are 4j, 1 [less than or equal to] j [less than or equal to] n - 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For example, the mean labelings of ([P.sub.7]; [S.sub.1]) and ([P.sub.6]; [S.sub.2]) are shown in Figure 7. ?

[FIGURE 7 OMITTED]

[section]3. Mean Graphs [[P.sub.m]; G]

Let G be a graph with fixed vertex v and let [[P.sub.m]; G] be the graph obtained from m copies of G by joining [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by means of an edge, for some j and 1 [less than or equal to] i [less than or equal to] m - 1.

For example [[P.sub.5]; [C.sub.3]] is shown in Figure 8.

[FIGURE 8 OMITTED]

Theorem 3.1 [[P.sub.m]; [C.sub.n]] is a mean graph.

Proof Let [u.sub.1], [u.sub.2], ... , [u.sub.m] be the vertices of [P.sub.m]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the vertices of the ith copy of [C.sub.n], 1 [less than or equal to] i [less than or equal to] m and joining [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by means of an edge, for some j.

Case (i) n = 4t,t = 1, 2, 3,...

Define f : V([[P.sub.m]; [C.sub.n]]) [right arrow] {0,1, 2, ... ,q} by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edge [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The label of the edges of the cycle are (n + 1)i - 1, (n + 1)i - 2, ... ,(n + 1)i - n, 1 [less than or equal to] i [less than or equal to] m.

For example, the mean labeling of [[P.sub.4]; [C.sub.8]] is shown in Figure 9.

[FIGURE 9 OMITTED]

Case (ii) n = 4t + 1, t =1, 2, 3, ...

Define f : V([[P.sub.m]; [C.sub.n]]) [right arrow] {0,1, 2, ... ,q} by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edge [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The label of the edges of the cycle are (n + 1)i - 1, (n + 1)i - 2, ... ,(n + 1)i - n, 1 [less than or equal to] i [less than or equal to] m.

For example, the mean labeling of [[P.sub.6]; [C.sub.5]] is shown in Figure 10.

[FIGURE 10 OMITTED]

Case (iii) n = 4t + 2, t = 1, 2, 3, ...

Define f : V([[P.sub.m]; [C.sub.n]]) - {0,1, 2, ... ,q} by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edge [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The label of the edges of the cycle are (n + 1)i - 1, (n + 1)i - 2, ... ,(n + 1)i - n, 1 [less than or equal to] i [less than or equal to] m.

For example, the mean labeling of [[P.sub.5]; [C.sub.6]] is shown in Figure 11.

[FIGURE 11 OMITTED]

Case (iv) n = 4t - 1, t = 1, 2, 3, ...

Define f : V([[P.sub.m]; [C.sub.n]]) [right arrow] {0,1, 2, ... ,q} by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edge [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The label of the edges of the cycle are (n + 1)i - 1, (n + 1)i - 2, ... ,(n + 1)i - n, 1 [less than or equal to] i [less than or equal to] m.

For example, the mean labeling of [[P.sub.7]; [C.sub.3]] is shown in Figure 12.

[FIGURE 12 OMITTED]

Theorem 3.2 [[P.sub.m]; [Q.sub.3]] is a mean graph.

Proof For 1 [less than or equal to] j [less than or equal to] 8, let v,. be the vertices in the ith copy of [Q.sub.3], 1 [less than or equal to] i [less than or equal to] m. Then define f on V[[P.sub.m]; [Q.sub.3]] as follows:

When i is odd.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When i is even.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edge [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The label of the edges of the cube are 13i - 1,13i - 2, ... ,13i - 12,1 [less than or equal to] i [less than or equal to] m.

For example the mean labeling of [[P.sub.4]; [Q.sub.3]] is shown in Figure 13.

[FIGURE 13 OMITTED]

Theorem 3.3 [[P.sub.m]; [C.sup.(2).sub.n]] is a mean graph.

Proof Let [u.sub.1], [u.sub.2], ... ,[u.sub.m] be the vertices of [P.sub.m] and the vertices [u.sub.i], 1 [less than or equal to] i [less than or equal to] m is attached with the center vertex in the ith copy of [C.sup.(2).sub.n]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (center vertex in the ith copy of [C.sup.(2).sub.n]).

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for 1 [less than or equal to] i [less than or equal to] m, 2 [less than or equal to] j [less than or equal to] n be the remaining vertices in the ith copy of [C.sup.(2).sub.n].

Then define f on V[[P.sub.m], [C.sup.(2).sub.n]] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Label the vertices of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as follows:

Case (i) When n is odd

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Case (ii) When n is even

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The label of the edge [u.sub.i][u.sub.i+1] is (2n + 1)i, 1 [less than or equal to] i [less than or equal to] m - 1 and the label of the edges of [C.sup.(2).sub.n]) are (2n + 1)i - 1, (2n + 1)i - 2, ... ,(2n + 1)i - 2n for 1 [less than or equal to] i [less than or equal to] m.

For example the mean labelings of [[P.sub.4], [C.sup.(2).sub.6]] and [[P.sub.5], [C.sup.(2).sub.3])] are shown in Figure 14.

[FIGURE 14 OMITTED]

References

[1] R. Balakrishnan and N. Renganathan, A Text Book on Graph Theory, Springer Verlag, 2000.

[2] S. Somasundaram and R. Ponraj, Mean labelings of graphs, National Academy Science letter, 26 (2003), 210-213.

[3] S. Somasundaram and R. Ponraj, Non--existence of mean labeling for a wheel, Bulletin of pure and Applied Sciences, (Section E Maths & Statistics) 22E(2003), 103-111.

[4] S. Somasundaram and R. Ponraj, Some results on mean graphs, Pure and Applied Mathematika Sciences, 58(2003), 29-35.

[5] S. Somasundaram and R. Ponraj, On Mean graphs of order [less than or equal to] 5, Journal of Decision and Mathematical Sciences, 9(1-3) (2004), 48-58.

[6] R. Ponra J and D. Ramya, Super mean labeling of graphs, Reprint.

[7] R. Ponra J and D. Ramya, On super mean graphs of order [less than or equal to] 5, Bulletin of Pure and Applied Sciences, (Section E Maths and Statistics) 25E (2006), 143-148.

[8] R. Ponraj and S. Somasundaram, Further results on mean graphs, Proceedings of Sacoeference, August 2005. 443-448.

[9] R. Ponra j and S.Somasundaram, Mean labeling of graphs obtained by identifying two graphs, Journal of Discrete Mathematical Sciences and Cryptography, 11(2)(2008), 239-252.

[10] R. Vasuki and A. Nagarajan, Some results on super mean graphs, International Journal of Mathematical Combinatorics, 3(2009), 82-96.

(1) Received March 26, 2010. Accepted June 18, 2010.