Minimum Covering Distance Energy of a Graph 1 Introduction

Applied Mathematical Sciences, Vol. 7, 2013, no. 111, 5525 - 5536 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.38477

Minimum Covering Distance Energy of a Graph

M. R. Rajesh Kanna, B. N. Dharmendra

Department of Mathematics Maharani’s Science College for Women J. L. B. Road, Mysore - 570 005, India [email protected], [email protected]

R. Pradeep Kumar

Research and Development Centre Bharathiar University Coimbatore 641 046, India [email protected]

Copyright c 2013 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Recently Prof.Chandrashekar Adiga et al.[1] have deﬁned the minimum covering energy, EC (G) of a graph G which depends on its particular minimum cover C. Motivated by this paper, we introduced the concept of minimum covering distance energy ECd(G) of a graph G and computed minimum covering distance energies of a star graph, complete graph, crown graph, bipartite graph and cocktail graphs. Upper and lower bounds for ECd(G) are also established.

Mathematics Subject Classiﬁcation: 05C50, 05C69

Keywords: Minimum covering set, Minimum covering distance matrix, Minimum covering distance eigenvalues, Minimum covering distance energy of a graph

1 Introduction

The concept of energy of a graph was introduced by I. Gutman [10] in the year 1978. Let G be a graph with n vertices {v1,v2, ..., vn} and m edges. Let A =(aij) be the adjacency matrix of the 5526 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar

graph. The eigenvalues λ1,λ2, ··· ,λn of A, assumed in non increasing order, are the eigenvalues of the graph G.AsA is real symmetric, the eigenvalues of G are real with sum equal to zero. The energy E(G) of G is deﬁned to be the sum of the absolute values of the eigenvalues of G. n i.e., E(G)= |λi|. i=1 For details on the mathematical aspects of the theory of graph energy see the reviews[11] ,papers [6, 7, 12] and the references cited there in. The basic properties including various upper and lower bounds for energy of a graph have been established in [14, 18], and it has found remarkable chemical applications in the molecular orbital theory of conjugated molecules [8, 13]. The distance matrix of G is the square matrix of order n whose (i, j)-entry is the distance (= length of the shortest path) between the vertices vi and vj . Let ρ1,ρ2, ..., ρn be the eigenvalues n of the distance matrix of G. The distance energy DE is deﬁned by DE = DE(G):= |ρi| i=1 . Detailed studies on distance energy can be found in [5, 9, 15, 16, 21]. Further studies on covering energy,maximum degree energy, dominating energies can be found in [1, 2, 19, 20]and the references cited there in.

2 Deﬁnitions and examples:

2.1 THE MINIMUM COVERING ENERGY A GRAPH

Let G be a simple graph of order n with vertex set V = {v1,v2, ..., vn} and edge set E. A subset C of V is called a covering set of G if every edge of G is incident to at least one vertex of C.Any covering set with minimum cardinality is called a minimum covering set. Let C be a minimum covering set of a graph G. The minimum covering matrix of G is the n × n matrix deﬁned by AC (G):=(aij), ⎧ ⎨ 1ifvivj ∈ E a i j v ∈ C where ij = ⎩ 1if= and i 0 otherwise

The characteristic polynomial ofAC (G) is denoted by fn(G, λ)= det(λI − AC (G)). The minimum covering eigenvalues of the graph G are the eigenvalues of AC (G). Since AC (G)is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order n λ1 λ2 ··· λn. The minimum covering energy of G is then deﬁned as EC (G)= |λi|. i=1

2.2 THE MINIMUM COVERING DISTANCE ENERGY

Let G be a simple graph of order n with vertex set V = {v1,v2, ..., vn} and edge set E. Let C be a minimum covering set of a graph G. The minimum covering distance matrix of G is the n × n matrix deﬁned by ACd(G):=(dij), 1ifi = j and vi ∈ D where dij = d(vi,vj) otherwise Minimum covering distance energy of a graph 5527

The characteristic polynomial of ACd(G) is denoted by fn(G, ρ)=det(ρI − ACd(G)). The minimum covering eigenvalues of the graph G are the eigenvalues of ACd(G). Since ACd(G)is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order ρ1 ρ2 ··· ρn. The minimum covering distance energy of G is deﬁned as ECd(G): = n |ρi| i=1 Note that the trace of ACd(G)=|C|. In this paper, we are interested in studying mathematical aspects of the minimum covering distance energy of a graph. The application of minimum covering distance energy in other branches of science have to be investigated.

EXAMPLE 1: The possible minimum covering sets for the following graph G [Figure 1] are i) C1={v1,v2,v5} ii) C2={v2,v4,v5} iii) C3={v1,v3,v5}

vs1 sv2 vs3

s s s v4 v5 v6

FIGURE - 1 ⎛ ⎞ 112112 ⎜ ⎟ ⎜ 111212⎟ ⎜ ⎟ A G ⎜ 210212⎟ i) Cd1 ( )=⎜ ⎟ ⎜ 122012⎟ ⎝ 111111⎠ 222210 Characteristic equation is ρ6 − 3ρ5 − 33ρ4 − 50ρ3 +5ρ2 +21ρ − 5=0. Minimum covering distance eigenvalues are ρ1 ≈−2.4142,ρ2 ≈−2.2203,ρ3 ≈−1.0000,ρ4 ≈ 0.2837,ρ5 ≈ 0.4142,ρ6 ≈ 7.9366.

Minimum covering distance energy, ECd1 (G) ≈ 14.2691 ⎛ ⎞ 012112 ⎜ ⎟ ⎜ 111212⎟ ⎜ ⎟ A G ⎜ 210212⎟ ii) Cd2 ( )=⎜ ⎟ ⎜ 122112⎟ ⎝ 111111⎠ 222210 5528 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar

Characteristic equation is ρ6 − 3ρ5 − 33ρ4 − 53ρ3 − 6ρ2 +13ρ − 1=0. Minimum covering distance eigenvalues are ρ1 ≈−2.3348,ρ2 ≈−2.2587,ρ3 ≈−0.8188,ρ4 ≈ 0.0825,ρ5 ≈ 0.3520,ρ6 ≈ 7.9778

Minimum covering distance energy, ECd2 (G) ≈ 13.8246. ∴ Minimum covering distance energy depends on the covering set.

3 MINIMUM COVERING DISTANCE ENERGY OF SOME STANDARD GRAPHS

Deﬁnition 3.1. The Cocktail party graph is denoted by Kn×2,is a graph having the vertex set n V = {ui,vi} and the edge set E = {uiuj,vivj : i = j} {uivj,viuj :1≤ i

Theorem 3.1. The minimum covering distance energy of Cocktail party graph Kn×2 is 4n. n Proof. Let Kn×2 be the Cocktail party graph with vertex set V = {ui,vi}. The minimum i=1 n−1 covering set is C = {ui,vi}. Then i=1 ⎛ ⎞ 1211... 1111 ⎜ ⎟ ⎜ 2111... 1111⎟ ⎜ ⎟ ⎜ 1112... 1111⎟ ⎜ ⎟ ⎜ 1121... 1111⎟ ⎜ ⎟ A K ⎜ ...... ⎟ Cd( n×2)=⎜ ...... ⎟ ⎜ ⎟ ⎜ 1111... 1211⎟ ⎜ ⎟ ⎜ 1111... 2111⎟ ⎝ ⎠ 1111... 1102 1111... 1120 Characteristic equation is ( ρ − 1)n−2( ρ + 2)( ρ +1)n−1[ ρ2 − (2n +1)ρ +2]=0

Minimum covering distance eigenvalues are ρ = 1[(n − 2)times] , ρ = −2 [one time], √ (2n +1)± 4n2 +4n − 7 ρ = −1[(n − 1) times], ρ = [one time each] 2 Minimum covering distance energy, ECd(Kn×2) √ (2n +1)+ 4n2 +4n − 7 = |1|(n − 2) + |−2|(1) + |−1|(n − 1) + + √ 2 (2n +1)− 4n2 +4n − 7 2 =4n.

Theorem 3.2. For n ≥ 3, the minimum covering distance energy of Star graph K1,n−1 is equal to 4n − 7. Minimum covering distance energy of a graph 5529

Proof. Consider the Star graph K1,n−1 with vertex set V = {v0,v1,v2, ..., vn−1}. The Minimum covering set C = {v0}. Then

⎛ ⎞ 111... 1 ⎜ ⎟ ⎜ 102... 2 ⎟ ⎜ ... ⎟ ACd(K1,n−1)=⎜ 120 2 ⎟ ⎜ . . . . . ⎟ ⎝ ...... ⎠ ... 122 0 n×n

Characteristic equation is (ρ +2)n−2[ ρ2 − (2n − 3)ρ +(n − 3)] = 0

The minimum covering distance eigenvalues are√ (2n − 3) ± 4n2 − 16n +21 ρ = −2[(n − 2) times], ρ = [one time each]. 2 E K Minimum covering distance energy√ is, Cd( 1,n−1) √ (2n − 3) + 4n2 − 16n +21 (2n − 3) − 4n2 − 16n +21 = |−2|(n − 2) + + 2 2 =4n − 7.

0 Deﬁnition 3.2. The Crown graph Sn for an integer n ≥ 2 is the graph with vertex set 0 {u1,u2, ..., un,v1,v2, ..., vn} and edge set {uivj :1≤ i, j ≤ n, i = j}. ∴ Sn coincides with the Complete bipartite graph Kn,n with horizontal edges removed.

0 Theorem 3.3.√ The minimum covering distance energy of the Crown graph Sn is equal to if n √ 5 √ =2 if n 101 + 2 17√ =3 (4n − 3) + (n − 1) 17 if n > 3

0 Proof. For the Crown graph Sn with vertex set V = {u1,u2, ..., un,v1,v2, ..., vn}. The minimum covering set is C = {u1,u2, ..., un} .Then

⎛ ⎞ 122... 2311... 1 ⎜ ⎟ ⎜ 212... 2131... 1 ⎟ ⎜ ⎟ ⎜ 221... 2113... 1 ⎟ ⎜ ⎟ ⎜ ...... ⎟ ⎜ ...... ⎟ ⎜ ⎟ 0 ⎜ 222... 1111... 3 ⎟ ACd(Sn)=⎜ ⎟ ⎜ 311... 1022... 2 ⎟ ⎜ ⎟ ⎜ 131... 1202... 2 ⎟ ⎜ ⎟ ⎜ 113... 1220... 2 ⎟ ⎜ ...... ⎟ ⎝ ...... ⎠ ...... 111 3222 0 (2n×2n) 5530 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar

2 2 i) Characteristic equation for n = 2 is (ρ − ρ − 1) =0. √ 1 ± 5 Minimum covering distance eigenvalues for n = 2 are ρ = [two √ 2 0 times each]. ∴ ECd(Sn)= 5.

ii) Characteristic equation for n ≥ 3is

[ ρ2 − (4n − 3)ρ +(3n2 − 10n − 2)](ρ2 +3ρ − 2)n−1 =0

Minimum covering√ distance eigenvalues for n ≥ 3 are √ (4n − 3) ± 4n2 +16n +17 −3 ± 17 ρ = [one time each], ρ = [(n − 1) times 2 2 each] 0 Minimum covering distance energy for n ≥ 3, ECd(Sn) √ √ √ (4n − 3) + 4n2 +16n +17 (4n − 3) − 4n2 +16n +17 −3+ 17 = + + 2 √ 2 2 −3 − 17 (n − 1) + (n − 1) 2 √ √ 0 Case 1 :Ifn = 3 then ECd(Sn)= 101 + 2 17. √ 0 Case 2 :Ifn>3 then ECd(Sn)=(4n − 3) + (n − 1) 17.

Theorem 3.4. The minimum covering distance energy of a Complete bipartite graph Km,n, (m ≤ n) is 3m +4n − 6.

Proof. For Complete bipartite graph Km,n(m ≤ n) with vertex set V = {u1,u2, ..., um,v1,v2, ..., vn} the minimum covering set C = {u1,u2, ..., um}. Then

⎛ ⎞ 122... 2111... 1 ⎜ ⎟ ⎜ 212... 2111... 1 ⎟ ⎜ ⎟ ⎜ 221... 2111... 1 ⎟ ⎜ ⎟ ⎜ ...... ⎟ ⎜ ...... ⎟ ⎜ ⎟ ⎜ 222... 1111... 1 ⎟ ACd(Km,n)=⎜ ⎟ ⎜ 111... 1022... 2 ⎟ ⎜ ⎟ ⎜ 111... 1202... 2 ⎟ ⎜ ⎟ ⎜ 111... 1220... 2 ⎟ ⎜ ...... ⎟ ⎝ ...... ⎠ ...... 111 1222 0 (m+n)×(m+n)

Characteristic equation is

(ρ +1)m−1(ρ +2)n−1[ ρ2 − (2m +2n − 3) − (3mn − 4m − 2n + 2)] = 0

Minimum covering distance eigenvalues are ρ = −1[(m − 1) times], ρ = −2 Minimum covering distance energy of a graph 5531

(2m +2n − 3) ± (4mn)2 − 4mn +4m +4n2 − 4n +1 [(n − 1) times] and ρ = 2 [one time each]

Minimum covering distance energy is, ECd(Km,n)= (2m +2n − 3) + (4mn)2 − 4mn +4m +4n2 − 4n +1 |−1|(m − 1) + |−2|(n − 1) + + 2 (2m +2n − 3) − (4mn)2 − 4mn +4m +4n2 − 4n +1 2 =(m − 1) + (2n − 2) + (2m +2n − 3) =3m +4n − 6

Theorem 3.5. For n ≥ 2 ,the minimum covering distance energy of Complete graph Kn is (n + 3)(n − 1).

Proof. For complete graphs the minimum covering distance matrix is same as minimum covering matrix [1], therefore the Minimum covering distance energy is equal to Minimum covering energy.

4 PROPERTIES OF MINIMUM COVERING EIGEN VALUES

Theorem 4.1. Let G be a simple graph with vertex set V = {v1,v2, ..., vn},edge set E and C = {u1,u2, ..., uk} be a minimum covering set.If ρ1,ρ2, ..., ρn are the eigenvalues of minimum n covering distance matrix ACd(G) then (i) ρi = |C| i=1 n 2 2 (ii) ρi =2m +2M + |C| where M = d(vi,vj) .

i=1 i

Proof. (i) We know that the sum of the eigenvalues of ACd(G) is the trace of ACd(G) n n ∴ ρi = dii = |C| . i=1 i=1 2 (ii) Similarly the sum of squares of the eigenvalues of ACd(G) is trace of [ACd(G)] 5532 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar

n n n 2 ∴ ρi = dijdji i=1 i=1 j=1 n 2 = (dii) + dijdji i=1 i=j n 2 2 = (dii) +2 (dij) i=1 i

Corollary 4.1. Let G be a (n,m)simple graph with diameter 2 and C = {u1,u2, ..., uk} be a minimum covering set.If ρ1,ρ2, ..., ρn are the eigenvalues of minimum covering distance matrix n 2 2 ACd(G) then ρi = |C| + 2(2n − 2n − 3m) . i=1

Proof. We know that in ACd(G) there are 2m elements with 1 and n(n − 1) − 2m elements with 2 and hence corollary follows from the above theorem.

5 BOUNDS FOR MINIMUM COVERING ENERGY

n n n 2 2 2 Cauchy Schwarz inequality is aibi ≤ ai bi i=1 i=1 i=1 n n n 2 2 If ai =1,bi =| ρi | then | ρi | ≤ 1 ρi i=1 i=1 i=1 2 [ECd(G)] ≤ n(2m +2M + |C|) [Theorem 4.1] =⇒ ECd(G) ≤ n(2m +2M + |C|) Minimum covering distance energy of a graph 5533

Since arithmetic mean is not smaller than geometric mean we have

1 1 | ρ || ρ |≥ | ρ || ρ | n(n − 1) n n − i j i j ( 1) i=j i=j 1 n 2(n−1) n n − = | ρi | ( 1) i=1 n 2 n = | ρi | i=1 n 2 n = ρi i=1 2 2 = |detACd(G)| n = P n

2 ∴ | ρi || ρj |≥n(n − 1)P n (5.1) i=j

n 2 2 Now consider, [ECd(G)] = | ρi | i=1 n 2 = | ρi | + | ρi || ρj | i=1 i=j 2 2 ∴ [ECd(G)] ≥ (|C| +2m +2M)+n(n − 1)P n [From (5.1)] 2 i.e., ECd(G) ≥ (|C| +2m +2M)+n(n − 1)P n

Theorem 5.2. If ρ1(G) is the largest minimum covering distance eigen value of ACd(G),then W G |C| ρ G ≥ 2 ( )+ |C| 1( ) n where is the cardinality of minimum covering set and W(G) is the Wiener index of G. XA X X ρ A Cd ∴ Proof. Let be any nonzero vector .Then by [3] ,We have 1( Cd) = max X=0 X X 2 d(vi,vj)+|C| JA J W G |C| ρ A ≥ Cd i

Lemma 5.1. Let G be a graph of diameter 2 and ρ1(G) is the largest minimum covering dis- n2 − m − n |C| A G ρ G ≥ 2 2 2 + |C| tance eigen value of Cd( ),then 1( ) n where is the cardinality of 5534 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar minimum covering set .

Proof. Let G be a connected graph of diameter 2 and di denotes the degree of vertex vi. Clearly i-th row of Add consists of di one’s and n − di − 1 two’s. By using Raleigh’s principle ,for J =[1, 1, 1, ··· , 1] we have n n [di × 1+(n − di − 1)2] + |C| [2n − di − 2] + |C| JA J n2 − m − n |C| ρ A ≥ Cd i=1 i=1 2 2 2 + 1( Cd) JJ = n = n = n .

Similar to Koolen and Moulton’s [17] upper bound for energy of a graph,upper bound for ECd(G) is given in the following theorem.

|C| +2n2 − 2n − 2m Theorem 5.3. If G is a (m, n) graph with diameter 2 and ≥ 1 then n |C| n2 − n − m |C| n2 − n − m2 E G ≤ +2 2 2 n − |C| n2 − n − m − +2 2 2 Cd( ) n + ( 1) +4 4 6 n .

Proof.

n n n 2 2 2 Cauchy-Schwartz inequality is aibi ≤ ai bi i=2 i=2 i=2 n n n 2 2 Put ai =1,bi =| ρi | then | ρi | ≤ 1 ρi i=2 i=2 i=2 2 2 2 ⇒ [ECd(G) − ρ1] ≤ (n − 1)(|C| +4n − 4n − 6m − ρ ) 1 2 2 ⇒ ECd(G) ≤ ρ1 + (n − 1)(|C| +4n − 4n − 6m − ρ1)

Let f(x)=x + (n − 1)(|C| +4n2 − 4n − 6m − x2).

For decreasing function f (x) ≤ 0 x(n − 1) ⇒ 1 − ≤ 0 (n − 1)(|C| +4n2 − 4n − 6m − x2) |C| n2 − n − m ⇒ x ≥ +4 4 6 n Minimum covering distance energy of a graph 5535

|C| +4n2 − 4n − 6m ∴ f(x) is decreasing in , |C| +4n2 − 4n − 6m . n |C| n2 − n − m |C| n2 − n − m +2 2 2 ∈ +4 4 6 , |C| n2 − n − m Clearly, n n +4 4 6 |C| n2 − n − m |C| n2 − n − m |C| n2 − n − m +2 2 2 ≥ , +2 2 2 ≤ +2 2 2 ≤ ρ Since n 1 we have n n 1 |C| n2 − n − m ∴ f ρ ≤ f +2 2 2 [By lema 5.1] ( 1) n |C| n2 − n − m i.e., E G ≤ f ρ ≤ f +2 2 2 Cd( ) ( 1) n |C| +2n2 − 2n − 2m i.e., ECd(G) ≤ f n |C| n2 − n − m |C| n2 − n − m2 i.e., E G ≤ +2 2 2 n − |C| n2 − n − m − +2 2 2 . Cd( ) n + ( 1) +4 4 6 n

Bapat and S.pati [4]proved that if the graph energy is a rational number then it is an even integer.Similar result for minimum covering distance energy is given in the following theorem.

Lemma 5.2. Let G be a graph with a minimum covering set C.If the minimum covering distance energy ECd(G) is a rational number,then ECd(G) ≡|C |(mod 2).

Proof. Proof is similar to Theorem 3.7 of [1].

References

[1] C.Adiga, A.Bayad, I.Gutman, S.A.Srinivas,The minimum covering energy of a graph,Kragujevac J. Sci. 34 (2012) 3956.

[2] C.Adiga and M. Smitha ,On Maximum Degree Energy of a Graph Int. J. Contemp. Math. Sciences, Vol. 4, (2009), no. 8, 385 - 396

[3] R.B.Bapat, page No.32,Graphs and Matrices,Hindustan Book Agency,(2011).

[4] R.B.Bapat, S.Pati, Energy of a graph is never an odd integer.Bull. Kerala Math. Assoc. 1, 129 - 132 (2011)

[5] S. B. Bozkurt, A. D. G¨ung¨or, B. Zhou, Note on the distance energy of graphs, MATCH Commun. Math. Comput. Chem. 64 (2010) 129134.

[6] D.Cvetkovi´c, I.Gutman (eds.),Applications of Graph Spectra (Mathematical Institu- tion,Belgrade,2009)

[7] D. Cvetkovi´c, I.Gutman (eds.) Selected Topics on Applications of Graph Spectra, (Mathe- matical Institute Belgrade,2011) 5536 M. R. Rajesh Kanna, B. N. Dharmendra and R. Pradeep Kumar

[8] A.Graovac, I.Gutman, N.Trinajsti´c,Topological Approach to the Chemistry of Conjugated Molecules (Springer, Berlin,1977)

[9] A. D. G¨ung¨or, S. B. Bozkurt, On the distance spectral radius and distance energy of graphs, Lin. Multilin. Algebra 59 (2011) 365370.

[10] I.Gutman,The energy of a graph.Ber. Math-Statist. Sekt. Forschungsz.Graz 103, 1-22 (1978)

[11] I.Gutman, X. Li, J.Zhang,in Graph Energy,ed. by M.Dehmer, F.Emmert - Streib. Analysis of Complex Networks. From Biology to Linguistics (Wiley - VCH, Weinheim,2009),pp. 145 174.

[12] I.Gutman, in The energy of a graph: Old and New Results,ed.by A. Betten, A. Kohnert, R. Laue, A. Wassermann. Algebraic Combinatorics and Applications (Springer, Berlin,2001),pp. 196 - 211.

[13] I.Gutman, O.E. Polansky,Mathematical Concepts in Organic Chemistry (Springer, Berlin,1986)

[14] Huiqing Liu,Mei Lu and Feng Tian,Some upper bounds for the energy of graphs Journal of Mathematical Chemistry, Vol. 41, No.1, (2007).

[15] G. Indulal, Sharp bounds on the distance spectral radius and the distance energy of graphs, Lin. Algebra Appl. 430 (2009) 106113.

[16] G. Indulal, I. Gutman, A. Vijayakumar, On distance energy of graphs,MATCH Commun. Math. Comput. Chem. 60 (2008) 461472.

[17] J.H. Koolen, V. Moulton, Maximal energy graphs. Adv.Appl. Math. 26,47 - 52 (2001)

[18] B.J.McClelland, Properties of the latent roots of a matrix: The estimation of π-electron energies.J. Chem. Phys.54, 640 - 643 (1971)

[19] M.R.Rajesh Kanna, B. N. Dharmendra, Shashi R and Ramyashree R A, Maximum degree energy of certain mesh derived networks.International Journal of Computer Applications, 78 No.8 (2013) 38-44.

[20] M.R.Rajesh Kanna, B. N. Dharmendra, and G. Sridhara, Minimum Dominating Energy of a Graph. International Journal of Pure and Applied Mathematics, 85, No. 4 (2013) 707-718.

[21] H. S. Ramane, D. S. Revankar, I. Gutman, S. B. Rao, B. D. Acharya, H. B. Walikar, Bounds for the distance energy of a graph, Kragujevac J. Math. 31 (2008) 5968.

Received: August 15, 2013