Hindawi Publishing Corporation International Journal of Combinatorics Volume 2014, Article ID 241723, 12 pages http://dx.doi.org/10.1155/2014/241723

Research Article Betweenness Centrality in Some Classes of Graphs

Sunil Kumar Raghavan Unnithan, Balakrishnan Kannan, and Madambi Jathavedan

Department of Computer Applications, Cochin University of Science and Technology, Cochin 682022, India

Correspondence should be addressed to Sunil Kumar Raghavan Unnithan; [email protected]

Received 21 May 2014; Accepted 21 October 2014; Published 25 December 2014

Academic Editor: Chris A. Rodger

Copyright © 2014 Sunil Kumar Raghavan Unnithan et al. 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.

There are several centrality measures that have been introduced and studied for real-world networks. They account for thedifferent characteristics that permit them to be ranked in order of importance in the network. Betweenness centrality is a measure of the influence of a vertex over the flow of information between every pair of vertices under the assumption that information primarily flows over the shortest paths between them. In this paper we present betweenness centrality of some important classes of graphs.

1. Introduction vertex centrality is in the potential of a vertex for control of information flow in the network. Positions are viewed Betweenness centrality plays an important role in analysis of as structurally central to the degree to which they stand social networks [1, 2], computer networks [3], and many other between others and can therefore facilitate, impede, or bias types of network data models [4–9]. the transmission of messages. Freeman in his papers [5, 16] In the case of communication networks the distance from classified betweenness centrality into three measures. The other units is not the only important property of a unit. three measures include two indexes of vertex centrality—one What is more important is which units lie on the shortest based on counts and one on proportions—and one index of paths (geodesics) among pairs of other units. Such units overall network or graph centralization. have control over the flow of information in the network. Betweenness centrality is useful as a measure of the potential 2.1. Betweenness Centrality of a Vertex. Betweenness central- of a vertex for control of communication. Betweenness ity 𝐶𝐵(V) for a vertex V is defined as centrality [10–14] indicates the betweenness of a vertex in a network and it measures the extent to which a vertex lies on 𝜎 (V) 𝐶 (V) = ∑ 𝑠𝑡 , the shortest paths between pairs of other vertices. In many 𝐵 𝜎 (1) real-world situations it has quite a significant role. 𝑠≠V=𝑡̸ 𝑠𝑡 Determining betweenness is simple and straightforward where 𝜎𝑠𝑡 isthenumberofshortestpathswithvertices𝑠 when only one geodesic connects each pair of vertices, where and 𝑡 as their end vertices, while 𝜎𝑠𝑡(V) is the number of the intermediate vertices can completely control communi- those shortest paths that include vertex V [16]. High centrality cation between pairs of others. But when there are several scores indicate that a vertex lies on a considerable fraction of geodesics connecting a pair of vertices, the situation becomes shortest paths connecting pairs of vertices. morecomplicatedandthecontroloftheintermediatevertices gets fractionated. (i) Every pair of vertices in a connected graph provides avaluelyingin[0, 1] to the betweenness centrality of all other vertices. 2. Background (ii) If there is only one geodesic joining a particular pair The concept of betweenness centrality was first introduced of vertices, then that pair provides a betweenness by Bavelas in 1948 [15]. The importance of the concept of centrality 1 to each of its intermediate vertices and 2 International Journal of Combinatorics

zero to all other vertices. For example, in a path graph, 1 a pair of vertices provides a betweenness centrality 1 to each of its interior vertices and zero to the exterior vertices. A pair of adjacent vertices always provides zero to all others. 2 6 (iii) If there are 𝑘 geodesics of length 2 joining a pair of vertices, then that pair of vertices provides a betweenness centrality 1/𝑘 to each of the intermediate

vertices. 0 Freeman [16]provedthatthemaximumvaluetakenby 𝐶 (V) 𝐵 is achieved only by the central vertex in a star as the central vertex lies on the geodesic (which is unique) joining 3 5 every pair of other vertices. In a star 𝑆𝑛 with 𝑛 vertices, the betweenness centrality of the central vertex is therefore the number of such geodesics which is ( 𝑛−1 ). The betweenness 2 centrality of each pendant vertex is zero since no pendant 4 vertex lies in between any geodesic. Again it can be seen that Figure 1: Wheel graph 𝑊7. the betweenness centrality of any vertex in a 𝐾𝑛 is zero since no vertex lies in between any geodesic as every geodesic is of length 1. ∗ The index, 𝐶𝐵(𝐺), determines the degree to which 𝐶𝐵(V ) exceeds the centrality of all other vertices in 𝐺.Since𝐶𝐵(𝐺) 2.2. Relative Betweenness Centrality. The betweenness cen- is the ratio of an observed sum of differences to its maximum trality increases with the number of vertices in the network, value, it will vary between 0 and 1. 𝐶𝐵(𝐺) = 0 if and only if all so a normalized version is often considered with the centrality ∗ 𝐶𝐵(V𝑖) are equal, and 𝐶𝐵(𝐺) = 1 if and only if one vertex V values scaled to between 0 and 1. Betweenness centrality completely dominates the network with respect to centrality. canbenormalizedbydividing𝐶𝐵(V) by its maximum value. Freemanshowedthatallofthesemeasuresagreeinassigning Among all graphs of 𝑛 vertices the central vertex of a star 𝑛−1 the highest centrality index to the star graph and the lowest graph 𝑆𝑛 has the maximum value which is ( ).Therelative 2 tothecompletegraph(seeTable 1). betweenness centrality is therefore defined as In this paper we present the betweenness centrality 󸀠 𝐶𝐵 (V) 2𝐶𝐵 (V) 󸀠 measures in some important classes of graphs which are the 𝐶𝐵 (V) = = 0≤𝐶𝐵 (V) ≤1. Max 𝐶𝐵 (V) (𝑛−1)(𝑛−2) basic components of larger complex networks. (2) 3. Betweenness Centrality of 2.3. Betweenness Centrality of a Graph. The betweenness Some Classes of Graphs centrality of a graph measures the tendency of a single vertex to be more central than all other vertices in the graph. It 3.1. Betweenness Centrality of Vertices in Wheels is based on differences between the centrality of the most centralvertexandthatofallothers.Freeman[16] defined the Wheel Graph 𝑊𝑛.AWheelgraph𝑊𝑛 is obtained by joining betweenness centrality of a graph as the average difference a new vertex to every vertex in a cycle 𝐶𝑛−1.Itwasinvented between the measures of centrality of the most central vertex bytheeminentgraphtheoristW.T.Tutte.Awheelgraphon andthatofallothervertices. 7verticesisshowninFigure 1. The betweenness centrality of a graph 𝐺 is defined as 𝑛 ∗ Theorem 1. The betweenness centrality of a vertex V in a wheel ∑𝑖=1 [𝐶𝐵 (V )−𝐶𝐵 (V𝑖)] 𝐶 (𝐺) = , graph 𝑊𝑛, 𝑛>5,isgivenby 𝐵 𝑛 ∗ (3) Max ∑𝑖=1 [𝐶𝐵 (V ) −𝐶𝐵 (V𝑖)] ∗ (𝑛−1)(𝑛−5) where 𝐶𝐵(V ) is the largest value of 𝐶𝐵(V𝑖) for any vertex V𝑖 { ,𝑖𝑓V 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 V𝑒𝑟𝑡𝑒𝑥, 𝑛 ∗ 𝐶 (V) = 2 in the given graph 𝐺 and Max ∑ [𝐶𝐵(V )−𝐶𝐵(V𝑖)] is the 𝐵 {1 𝑖=1 { ,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. maximum possible sum of differences in centrality for any {2 graph of 𝑛 vertices which occur in star with the value 𝑛−1 𝑛−1 (5) times 𝐶𝐵(V) of the central vertex, that is, (𝑛 − 1) ( 2 ). 𝐺 Therefore the betweenness centrality of is defined as Proof. In the wheel graph 𝑊𝑛 the central vertex is adjacent 𝑛 ∗ to each vertex of the cycle 𝐶𝑛−1. Consider the central vertex 2∑𝑖=1 [𝐶𝐵 (V )−𝐶𝐵 (V𝑖)] 𝐶𝐵 (𝐺) = of 𝑊𝑛 for 𝑛>5.On𝐶𝑛−1 each pair of adjacent vertices con- (𝑛−1)2 (𝑛−2) tributes centrality 0,eachpairofalternateverticescontributes (4) 1/2 1 ∑𝑛 [𝐶󸀠 (V∗)−𝐶󸀠 (V )] centrality , and all other pairs contribute centrality to 𝑖=1 𝐵 𝐵 𝑖 𝑛−1 𝐶 or 𝐶𝐵 (𝐺) = . the central vertex. Since there are vertices on 𝑛−1, (𝑛−1) there exist 𝑛−1adjacent pairs, 𝑛−1alternate pairs, and International Journal of Combinatorics 3

Table 1: Graphs showing extreme betweenness.

󸀠 𝐺𝐶𝐵(V)𝐶𝐵 (V)𝐶𝐵(𝐺) { 𝑛−1 {( ) for central vertex {1 for central vertex 𝑆𝑛 { 2 { 1 { {0 for other vertices {0 for other vertices

𝐾𝑛 000

𝑛−1 ( 2 ) − 2(𝑛 − 1) = (𝑛 − 1)(𝑛 −6)/2 other pairs. Therefore 1 the betweenness centrality of the central vertex is given by (1/2)(𝑛−1)+1((𝑛−1)(𝑛−6)/2) = (𝑛−1)(𝑛−5)/2.Nowforany vertex on 𝐶𝑛−1, there are two geodesics joining its adjacent 𝐶 vertices on 𝑛−1, one of which passing through it. Therefore 2 6 the betweenness centrality of any vertex on 𝐶𝑛−1 is 1/2.

Note.Itcanbeseeneasilythat𝐶𝐵(V)=0for every vertex V in 𝑊4 and 2 { , V {3 if is the central vertex, 3 5 𝐶 (V) = 𝐵 {1 (6) { , {3 otherwise

4 in 𝑊5. The relative centrality and graph centrality are as follows: Figure 2: Complete graph 𝐾6 −(V1, V4). 2𝐶 (V) 𝐶󸀠 (V) = 𝐵 𝐵 (𝑛−1)(𝑛−2) The relative centrality and graph centrality are as follows: (𝑛−5) { , if V is the central vertex, 2𝐶 (V) 𝑛−2 𝐶󸀠 (V) = 𝐵 = { 1 𝐵 { , , (𝑛−1)(𝑛−2) (𝑛−1)(𝑛−2) otherwise { 2 { , V ≠ V𝑖, V𝑗, 𝑛 󸀠 ∗ 󸀠 2 2 ∑𝑖=1 [𝐶𝐵 (V )−𝐶𝐵 (V𝑖)] 𝑛 −6𝑛+4 = {(𝑛−1)(𝑛−2) 𝐶 (𝑊 )= = . (9) 𝐵 𝑛 (𝑛−1) (𝑛−1)(𝑛−2) {0, otherwise, (7) 𝑛 󸀠 ∗ 󸀠 ∑ [𝐶 (V )−𝐶 (V )] 4 𝐶 (𝐺) = 𝑖=1 𝐵 𝐵 𝑖 = . 𝐵 (𝑛−1) (𝑛−1)2 (𝑛−2)2 3.2. Betweenness Centrality of Vertices in the Graph 𝐾𝑛 −𝑒. A complete graph on 6 vertices with one edge deleted is shown in Figure 2. 3.3. Betweenness Centrality of Vertices in Complete Bipartite Graphs Theorem 2. Let 𝐾𝑛 be a complete graph on 𝑛 vertices and 𝑒= (V𝑖, V𝑗) an edge of it. Then the betweenness centrality of vertices Complete Bipartite Graphs 𝐾𝑚,𝑛.Agraphiscompletebipartite in 𝐾𝑛 −𝑒is given by if its vertices can be partitioned into two disjoint nonempty sets 𝑉1 and 𝑉2 such that two vertices 𝑥 and 𝑦 are adjacent if 1 { and only if 𝑥∈𝑉1 and 𝑦∈𝑉2.If|𝑉1|=𝑚and |𝑉2|=𝑛,such ,𝑖𝑓V ≠ V𝑖, V𝑗, 𝐶𝐵 (V) = {𝑛−2 (8) a graph is denoted 𝐾𝑚,𝑛. For example, see 𝐾3,4 in Figure 3. {0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. Theorem 3. The betweenness centrality of a vertex ina Proof. Suppose the edge (V𝑖, V𝑗) is removed from 𝐾𝑛.Now complete 𝐾𝑚,𝑛 is given by V𝑖 and V𝑗 can be joined by means of any of the remaining 𝑛−2vertices. Thus there are 𝑛−2geodesics joining V𝑖 and { 1 𝑛 { ( ), 𝑖𝑓 deg (V) =𝑛, V𝑗 each containing exactly one vertex as intermediary. This {𝑚 2 1/(𝑛−2) 𝑛−2 𝐶 (V) = provides a betweenness centrality to each of the 𝐵 { 1 𝑚 (10) V V { ( ), 𝑖𝑓 (V) =𝑚. vertices. Again 𝑖 and 𝑗 do not lie in between any geodesics 𝑛 2 deg and therefore their betweenness centralities are zero. { 4 International Journal of Combinatorics

u1 u2 u3 1

2 6

1 2 3 4

Figure 3: 𝐾3,4. 3 5

Proof. Consider a complete bipartite graph 𝐾𝑚,𝑛 with a bipartition {𝑈, 𝑊} where 𝑈={𝑢1,𝑢2,...,𝑢𝑚} and 𝑊= 4 {𝑤 ,𝑤 ,...,𝑤 } 𝑈 1 2 𝑛 . The distance between any two vertices in (3) (or in 𝑊) is 2. Consider a vertex 𝑢∈𝑈. Now any pair of Figure 4: Cocktail party graph CP . vertices in 𝑊 contributes a betweenness centrality 1/𝑚 to 𝑢. 𝑛 𝑛 Since there are ( 2 ) pairsofverticesin𝑊, 𝐶𝐵(𝑢) = (1/𝑚) ( 2 ). In a similar way it can be shown that, for any vertex 𝑤 in 𝑊, Theorem 4. The betweenness centrality of each vertex ofa 𝑚 𝐶𝐵(𝑤) = (1/𝑛) ( 2 ). cocktail party graph of order 2𝑛 is 1/2.

Proof. Let the cocktail party graph CP(𝑛) be obtained The relative centrality and graph centrality are as follows: from the complete graph 𝐾2𝑛 with vertices {V1,..., V𝑛, V𝑛+1,...,V2𝑛} by deleting a perfect matching {(V1, V𝑛+1), 󸀠 2𝐶 (V) 𝐵 (V2, V𝑛+2),...,(V𝑛, V2𝑛)}. Now for each pair (V𝑖, V𝑛+𝑖) there is a 𝐶𝐵 (V) = (𝑚+𝑛−1)(𝑚+𝑛−2) geodesic of length 2 passing through each of the other 2𝑛−2 2 vertices. Thus for any particular vertex, there are 𝑛−1pairs { of vertices of the above matching not containing that vertex {(𝑚+𝑛−1)(𝑚+𝑛−2) { giving a betweenness centrality 1/(2𝑛 − 2) to that vertex. { { 1 𝑛 Therefore the betweenness centrality of any vertex is given { × ( ), (V) =𝑛, { 𝑚 2 if deg by (𝑛 − 1)/(2𝑛 − 2) .= 1/2 = { 2 { { The relative centrality and graph centrality are as follows: {(𝑚+𝑛−1)(𝑚+𝑛−2) { { 1 𝑚 󸀠 2𝐶 (V) 1 { × ( ), (V) =𝑚, 𝐶 (V) = 𝐵 = , if deg 𝐵 (2𝑛 − 1)(2𝑛 − 2) (2𝑛 − 1)(2𝑛 − 2) { 𝑛 2 (12) 𝐶 (𝐺) =0. ∑𝑛 [𝐶󸀠 (V∗)−𝐶󸀠 (V )] 𝐵 𝐶 (𝐾 )= 𝑖=1 𝐵 𝐵 𝑖 𝐵 𝑚,𝑛 (𝑚+𝑛−1) 3.5. Betweenness Centrality of Vertices in Crown Graphs. The 𝑚3 −𝑛3 −(𝑚2 −𝑛2) crown graph [18] is the unique 𝑛−1regular graph with 2𝑛 { { , if 𝑚>𝑛, vertices obtained from a complete bipartite graph 𝐾𝑛,𝑛 by {𝑛 𝑚+𝑛−1 2 𝑚+𝑛−2 = ( ) ( ) deleting a perfect matching (see Figure 5). A crown graph on { 2 2 { 𝑛 (𝑛−1) −𝑚 (𝑚−1) 2𝑛 vertices can be viewed as an undirected graph with two sets { , if 𝑛>𝑚. 𝑢 V 𝑢 V 𝑚 (𝑚+𝑛−1)2 (𝑚+𝑛−2) of vertices 𝑖 and 𝑖 andwithanedgefrom 𝑖 to 𝑗 whenever { 𝑖 =𝑗̸ 𝐿 (11) . It is the graph complement of the ladder graph 2𝑛.The crown graph is a distance-transitive graph.

3.4. Betweenness Centrality of Vertices in Cocktail Party Theorem 5. The betweenness centrality of each vertex ofa Graphs. The cocktail party graph CP(𝑛) [17] is a unique crown graph of order 2𝑛 is (𝑛 + 1)/2. regular graph of degree 2𝑛 − 2 on 2𝑛 vertices. It is obtained from 𝐾2𝑛 by deleting a perfect matching (see Figure 4). Proof. Let the crown graph be the complete bipartite graph The cocktail party graph of order 𝑛 is a complete 𝑛-partite 𝐾𝑛,𝑛 with vertices {𝑢1,...,𝑢𝑛, V1,...,V𝑛} minus a perfect graph with 2 vertices in each partition set. It is the graph matching {(𝑢1, V1), (𝑢2, V2),...,(𝑢𝑛, V𝑛)}. Consider any vertex; 󸀠 complement of the ladder rung graph 𝐿𝑛 which is the graph say 𝑢1. Now for each pair (𝑢𝑖, V𝑖) other than (𝑢1, V1) there are union of 𝑛 copies of the path graph 𝑃2 and the dual graph of 𝑛−2paths of length 3 passing through 𝑢1 out of (𝑛 − 1)(𝑛 − 2) the hypercube 𝑄𝑛 [18]. paths joining 𝑢𝑖 and V𝑖.Sincethereare𝑛−1such pairs, they International Journal of Combinatorics 5

u u u u 1 2 3 4 Corollary 7. Graph centrality of 𝑃𝑛 is given by

𝑛 (𝑛+1) { , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑, {6 (𝑛−1)(𝑛−2) 𝐶𝐵 (𝑃𝑛)={ (16) { 𝑛 (𝑛+2) { ,𝑖𝑓𝑛𝑖𝑠𝑒V𝑒𝑛. {6 (𝑛−1)2

1 2 3 4

Figure 5: Crown graph with 8 vertices. Proof. When 𝑛 is even, by definition

give V1 the betweenness centrality (𝑛 − 1) × ((𝑛 − 2)/(𝑛 − 2 𝑛 𝐶 (𝑃 )= ∑ [𝐶 (V∗)−𝐶 (V )] 1)(𝑛 − 2)) =. 1 Again for each pair from {V2, V3, V4,...,V𝑛} 𝐵 𝑛 2 𝐵 𝐵 𝑖 (𝑛−1) (𝑛−2) 𝑖=1 there exists exactly one path passing through V1 out of 𝑛− 𝑛−1 2 paths. Since there are ( 2 ) such pairs, they give V1 the 4 𝑛−1 = betweenness centrality ( 2 ) (1/(𝑛−2)) = (𝑛−1)/2. Therefore (𝑛−1)2 (𝑛−2) V 1 + (𝑛 − 1)/2 = the betweenness centrality of 1 is given by 𝑛 (𝑛−2) 𝑛 (𝑛−2) (𝑛+1)/2. Since the graph is vertex transitive, the betweenness ×{[ −0]+[ −1⋅(𝑛−2)] centrality of any vertex is given by (𝑛 + 1)/2. 4 4 𝑛 (𝑛−2) 𝑛−4 𝑛−2 +⋅⋅⋅+[ −( )(𝑛− )]} 4 2 2 The relative centrality and graph centrality are as follows: 4 = 2𝐶 (V) 𝑛+1 (𝑛−1)2 (𝑛−2) 𝐶󸀠 (V) = 𝐵 = , 𝐵 (2𝑛 − 1)(2𝑛 − 2) (2𝑛 − 1)(2𝑛 − 2) 𝑛 (𝑛−2) 𝑛−2 (13) ×{[ × ] 4 2 𝐶𝐵 (𝐺) =0. −[1(𝑛−2) +2(𝑛−3) 3.6. Betweenness Centrality of Vertices in Paths 𝑛−4 𝑛−2 +⋅⋅⋅+( )(𝑛− )]} Theorem 6. The betweenness centrality of any vertex in a path 2 2 4 graph is the product of the number of vertices on either side of = that vertex in the path. (𝑛−1)2 (𝑛−2)

2 (𝑛−4)/2 (𝑛−4)/2 Proof. Consider a path graph 𝑃𝑛 of 𝑛 vertices {V1, V2,...,V𝑛} 𝑛 (𝑛−2) ×{ −𝑛 ∑ 𝑘+ ∑ 𝑘 (𝑘+1)} (see Figure 6). Take a vertex V𝑘 in 𝑃𝑛. Then there are 𝑘−1 8 𝑘=1 𝑘=1 vertices on one side and 𝑛−𝑘vertices on the other side of V𝑘. 4 Consequently there are (𝑘−1)×(𝑛−𝑘)number of geodesics = 2 containing V𝑘.Hence𝐶𝐵(V𝑘) = (𝑘 − 1)(𝑛 − 𝑘). (𝑛−1) (𝑛−2) 𝑛 (𝑛−2)2 𝑛 (𝑛−2)(𝑛−4) ×{ − Note that, by symmetry, vertices at equal distance away 8 8 from both ends of 𝑃𝑛 havethesamecentralityanditis (𝑛−2)(𝑛−4) (𝑛−2)(𝑛−3)(𝑛−4) maximumatthecentralvertex(vertices)andminimumatthe + + } end vertices. Consider 8 24 1 𝑛 (𝑛−2) = { , when 𝑛 is even, (𝑛−1)2 𝐶 (V )= 4 Max 𝐵 𝑘 {(𝑛−1)2 (14) 𝑛 (𝑛−2) 𝑛 (𝑛−4) , 𝑛 . ×{ − { 4 when is odd 2 2 (𝑛−4) (𝑛−3)(𝑛−4) Relative centrality of any vertex V𝑘 is given by + + } 2 6

󸀠 2𝐶𝐵 (V𝑘) 2 (𝑘−1)(𝑛−𝑘) 𝑛 (𝑛+2) 𝐶 (V ) = = . (15) = . 𝐵 𝑘 (𝑛−1)(𝑛−2) (𝑛−1)(𝑛−2) 6 (𝑛−1)2 (17) 6 International Journal of Combinatorics

1 2 3 4 k−1 k k+1 n−1 n

Figure 6: Path graph 𝑃𝑛.

When 𝑛 is odd, by definition 1 2 3 4 5 2 𝑛 𝐶 (𝑃 )= ∑ [𝐶 (V∗)−𝐶 (V )] 𝐵 𝑛 2 𝐵 𝐵 𝑖 (𝑛−1) (𝑛−2) 𝑖=1 4 = (𝑛−1)2 (𝑛−2) (𝑛−1)2 (𝑛−1)2 6 7 8 9 10 ×{[ −0]+[ −1⋅(𝑛−2)] 4 4 Figure 7: Ladder graph 𝐿5. (𝑛−1)2 𝑛−3 𝑛−1 +⋅⋅⋅+[ −( )(𝑛− )]} 4 2 2 4 𝑃 ×𝑃 2𝑛 = 2 𝑛 (see Figure 7). It is a planar undirected graph with (𝑛−1)2 (𝑛−2) vertices and 𝑛 + 2(𝑛 −1) edges. (𝑛−1)2 𝑛−1 Theorem 8. ×{[ × ] The betweenness centrality of a vertex in a ladder 4 2 graph 𝐿𝑛 is given by

−[1(𝑛−2) +2(𝑛−3) 𝑘−1 𝑛−𝑘 𝑘−𝑗 𝐶 (V )=(𝑘−1)(𝑛−𝑘) + ∑ ∑ 𝑛−3 𝑛−1 𝐵 𝑘 𝑘−𝑗+𝑖 +⋅⋅⋅+( )(𝑛− )]} 𝑗=0 𝑖=1 2 2 (19) 𝑘−2 𝑛−𝑘 𝑖+1 4 + ∑ ∑ , 1≤𝑘≤𝑛. = 𝑘−𝑗+𝑖 (𝑛−1)2 (𝑛−2) 𝑗=0 𝑖=0

(𝑛−1)3 (𝑛−3)/2 (𝑛−3)/2 ×{ −𝑛 ∑ 𝑘+ ∑ 𝑘 (𝑘+1)} 8 𝑘=1 𝑘=1 Proof. By symmetry, let V𝑘 be any vertex such that 1≤𝑘≤ (𝑛 + 1)/2 4 . Consider the paths (in Figure 8)fromupperleft = {V1,...,V𝑘−1} {V𝑘+1,...,V𝑛} 2 vertices to upper right vertices (𝑛−1) (𝑛−2) which give the betweenness centrality (𝑛−1)3 𝑛 (𝑛−1)(𝑛−3) ×{ − 8 8 (𝑘−1)(𝑛−𝑘) . (20) (𝑛−1)(𝑛+1)(𝑛−3) + } 24 Now consider the paths from lower left vertices {V𝑛+1, 1 = ...,V𝑛+𝑘} to the upper right vertices {V𝑘+1,...,V𝑛} of V𝑘 which (𝑛−1)(𝑛−2) give the betweenness centrality (𝑛−1)2 𝑛 (𝑛−3) (𝑛+1)(𝑛−3) ×{ − + } 2 2 6 1 1 1 𝑘{ + +⋅⋅⋅+ } } 𝑛 (𝑛+1) 𝑘+1 𝑘+2 𝑛 } = . } 6 (𝑛−1)(𝑛−2) 1 1 1 } (𝑘−1) { + +⋅⋅⋅+ } } 𝑘−1 𝑛−𝑘 𝑘−𝑗 (18) 𝑘 𝑘+1 𝑛−1 } = ∑ ∑ . . } 𝑘−𝑗+𝑖 . } 𝑗=0 𝑖=1 } 1 1 1 } { + +⋅⋅⋅+ } } 2 3 𝑛−(𝑘−1) 3.7. Betweenness Centrality of Vertices in Ladder Graphs. The } ladder graph 𝐿𝑛 [19, 20] is defined as the Cartesian product (21) International Journal of Combinatorics 7

1 2 3 k−1 k k+1 n−2 n−1 n

n+1 n+2 n+3 n+k−1 n+k n+k+1 2n−2 2n−1 2n

Figure 8: Ladder graph 𝐿𝑛.

2 3 4 5 k−3 k−2 k−1 k

1 k+1

2k 2k−1 2k−2 2k−3 k+5 k+4 k+3 k+2

Figure 9: Even cycle with 2𝑘 vertices.

Again consider the paths from upper left vertices V2𝑘 to V𝑘+1+𝑖 passing through V1 for 𝑖 = 2,3,...,𝑘 −1 and {V ,...,V } {V ,...,V } V 𝑘−1 1 𝑘−1 to the lower right vertices 𝑛+𝑘 2𝑛 of 𝑘 each contributes centrality 1 to V1 giving a total ∑𝑖=2 (𝑘−𝑖)= which give the betweenness centrality (𝑘 − 1)(𝑘 − 2)/2. Therefore the betweenness centrality of V1 is (1/2)(𝑘 − 1) + (𝑘 − 1)(𝑘 − 2)/2 = (1/2)(𝑘 −1)2 = (1/8)(𝑛 − 2)2 1 2 𝑛−(𝑘−1) . { + +⋅⋅⋅+ } } 𝐶 } Since 𝑛 is vertex transitive, the betweenness centrality of any 𝑘 𝑘+1 𝑛 } (1/8)(𝑛 − 2)2 } vertex is given by . 1 2 𝑛−(𝑘−1) } 𝑘−2 𝑛−𝑘 { + +⋅⋅⋅+ } } 𝑖+1 + 𝑘−1 𝑘 𝑛−1 } = ∑ ∑ . Case 2 (when 𝑛 is odd). Let 𝑛=2𝑘+1, 𝑘∈Z , and let . } 𝑘−𝑗+𝑖 . } 𝑗=0 𝑖=0 𝐶𝑛 =(V1, V2,...,V2𝑘+1) be the given cycle. Consider a vertex . } } V1 (see Figure 10). Then V𝑘+1 and V𝑘+2 are its antipodal vertices 1 2 𝑛−(𝑘−1) } { + +⋅⋅⋅+ } at a distance 𝑘 from V1 and there is no geodesic path from 2 3 𝑛−(𝑘−2) } the vertexes V𝑘+1 and V𝑘+2 to any other vertex passing through (22) V1.HenceweomitV1 and the pair (V𝑘+1, V𝑘+2). Now consider paths of length ≤𝑘passing through V1.Thereare𝑘+1−𝑖paths The above three equations when combined get the result. joining V𝑖 to vertices from V2𝑘+1 to V𝑘+1+𝑖 passing through V1 for 𝑖 = 2,3,...,𝑘 and each contributes a betweenness 𝑘 centrality 1 to V1 giving a total of ∑𝑖=2(𝑘+1−𝑖)= 𝑘(𝑘−1)/2= 3.8. Betweenness Centrality of Vertices in Cycles (𝑛 − 1)(𝑛 − 3)/8.Since𝐶𝑛 is vertex transitive, the betweenness centrality of any vertex is given by (𝑛 − 1)(𝑛 − 3)/8. Theorem 9. The betweenness centrality of a vertex in a cycle 𝐶𝑛 is given by

2 {(𝑛−2) The relative centrality and graph centrality are as follows: { ,𝑖𝑓𝑛𝑖𝑠𝑒V𝑒𝑛, 8 𝐶𝐵 (V) = { (23) {(𝑛−1)(𝑛−3) 𝐶 (V) 2𝐶 (V) , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑. 𝐶󸀠 (V) = 𝐵 = 𝐵 { 8 𝐵 Max 𝐶𝐵 (V) (𝑛−1)(𝑛−2) Proof. 𝑛−2 + { , if 𝑛 is even, Case 1 (when 𝑛 is even). Let 𝑛=2𝑘, 𝑘∈Z , and let = 4 (𝑛−1) (24) { 𝑛−3 𝐶𝑛 =(V1, V2,...,V2𝑘) be the given cycle. Consider a vertex , 𝑛 , {4 (𝑛−2) if is odd V1 (see Figure 9). Then V𝑘+1 is its antipodal vertex and there is no geodesic from V𝑘+1 to any other vertex passing through 𝐶𝐵 (𝐶𝑛)=0. V1.Henceweomitthepair(V1, V𝑘+1). Consider other pairs of antipodal vertices (V𝑖, V𝑘+𝑖) for 𝑖 = 2,3,...,𝑘.Foreach pair of these antipodal vertices there exist two paths of the 3.9. Betweenness Centrality of Vertices in Circular Ladder same length 𝑘 and one of them contains V1.Thuseachpair Graphs 𝐶𝐿 𝑛. The circular ladder graph CL𝑛 consists of two contributes 1/2 to the centrality of V1 which gives a total of concentric 𝑛-cycles in which each pair of the 𝑛 corresponding (1/2)(𝑘 − 1). Now consider all paths of length less than 𝑘 vertices is joined by an edge (see Figure 11). It is a 3-regular containing V1.Thereare𝑘−𝑖paths joining V𝑖 to vertices from simple graph isomorphic to the Cartesian product 𝐾2 ×𝐶𝑛. 8 International Journal of Combinatorics

2 3 4 5 k−2 k−1 k k+1

1

2k+1 2k 2k−1 2k−2 k+5 k+4 k+3 k+2

Figure 10: Odd cycle with 2𝑘 + 1 vertices.

Again the pair (𝑢𝑘+1, V1) contributes to 𝑢1 the betweenness centrality 2/(2𝑘+2). Therefore the betweenness centrality of 𝑢1 is given as (𝑘−1)2 𝑘2 1 1 𝐶 (𝑢 )= + − + 𝐵 1 2 2 𝑘+1 𝑘+1 (𝑘−1)2 𝑘2 (2𝑘 − 1)2 +1 = + = (27) 2 2 4 (𝑛−1)2 +1 = . 4 + Case 2 (when 𝑛 is odd). Let 𝑛=2𝑘+1, 𝑘∈Z .Let 󸀠 𝐶2𝑘+1 =(𝑢1,𝑢2,...,𝑢2𝑘+1) be the outer cycle and 𝐶2𝑘+1 = Figure 11: Circular ladder. (V1, V2,...,V2𝑘+1) the inner cycle. Consider any vertex; say 𝑢1 in 𝐶2𝑘+1. Then its betweenness centrality as a vertex in 𝐶2𝑘+1 is 𝑘(𝑘 − 1)/2. Now consider the geodesics from outer vertices 𝑢𝑖 Theorem 10. The betweenness centrality of a vertex ina to the inner vertices V1, V2𝑘+1,...,V𝑘+𝑖+1 for 𝑖 = 2,...,𝑘+ 1 circular ladder 𝐶𝐿 𝑛 is given by (see Figure 13)andfrom𝑢2𝑘+3−𝑖 to V1, V2,...,V𝑘+2−𝑖 for 𝑖= 2 {(𝑛−1) +1 2,3,...,𝑘+1which give a betweenness centrality { , 𝑤ℎ𝑒𝑛𝑛𝑖𝑠𝑒V𝑒𝑛, 4 𝑘 𝐶𝐵 (V) = { 2 (25) 1 2 𝑘+2−𝑖 {(𝑛−1) 2∑ ( + +⋅⋅⋅+ ) , 𝑤ℎ𝑒𝑛𝑛𝑖𝑠𝑜𝑑𝑑. 𝑖 𝑖+1 𝑘+1 { 4 𝑖=2 1 1+2 1+2+⋅⋅⋅+𝑘 Proof. =2( + +⋅⋅⋅+ ) + 2 3 𝑘+1 Case 1 (when 𝑛 is even). Let 𝑛=2𝑘, 𝑘∈Z .Let𝐶2𝑘 = 󸀠 (𝑢1,𝑢2,...,𝑢2𝑘) be the outer cycle and 𝐶 =(V1, V2,...,V2𝑘) 2𝑘 𝑘+1 (1+2+⋅⋅⋅+𝑝−1) (28) the inner cycle. Consider any vertex; say 𝑢1 in 𝐶2𝑘.Then =2∑ 2 𝑝 its betweenness centrality as a vertex in 𝐶2𝑘 is (𝑘 − 1) /2. 𝑝=2 Now the geodesics from outer vertices 𝑢𝑖 to the inner vertices 𝑘+1 V1, V2𝑘,...,V𝑘+𝑖 for 𝑖 = 2,...,𝑘 (see Figure 12)andfrom 𝑝−1 𝑘 (𝑘+1) =2∑ = . 𝑢2𝑘+2−𝑖 to V1, V2,...,V𝑘+2−𝑖 for 𝑖 = 2,...,𝑘 by symmetry 𝑝=2 2 2 contribute to 𝑢1 the betweenness centrality Therefore the betweenness centrality of 𝑢1 is given as 𝑘 1 2 𝑘+1−𝑖 𝑘+2−𝑖 2∑ ( + +⋅⋅⋅+ + ) 𝑘 (𝑘−1) 𝑘 (𝑘+1) (𝑛−1)2 𝑖 𝑖+1 𝑘 2𝑘 + 2 𝐶 (𝑢 )= + =𝑘2 = . (29) 𝑖=2 𝐵 1 2 2 4 1 1+2 1+2+⋅⋅⋅+𝑘−1 =2( + +⋅⋅⋅+ 2 3 𝑘 The relative centrality and graph centrality are as follows: 2+3+⋅⋅⋅+𝑘 2𝐶 (V) + ) 𝐶󸀠 (V) = 𝐵 2𝑘 + 2 (26) 𝐵 (2𝑛 − 1)(2𝑛 − 2)

𝑘 2 (1+2+⋅⋅⋅+𝑝−1) 𝑘 (𝑘+1) −2 { (𝑛−1) +1 =2∑ + { , 𝑛 𝑝 2 (𝑘+1) 2 (2𝑛 − 1)(2𝑛 − 2) when is even, (30) 𝑝=2 = { { (𝑛−1) , when 𝑛 is odd, 𝑘 𝑝−1 𝑘 (𝑘+1) −2 𝑘2 1 {4 (2𝑛 − 1) =2∑ + = − . 2 2 (𝑘+1) 2 𝑘+1 𝑝=2 𝐶𝐵 (𝐺) =0. International Journal of Combinatorics 9

u2 u3 u4 u5 uk−3 uk−2 uk−1 uk

2 3 4 5 k−3 k−2 k−1 k u1 1 k+1 uk+1

2k 2k−1 2k−2 2k−3 k+5 k+4 k+3 k+2

u2k u2k−1 u2k−2 u2k−3 uk+5 uk+4 uk+3 uk+2

Figure 12: Circular ladder CL2𝑘.

u2 u3 u4 u5 uk−2 uk−1 uk uk+1

2 3 4 5 k−2 k−1 k k+1

u1 1

2k+1 2k 2k−1 2k−2 k+5 k+4 k+3 k+2

u2k+1 u2k u2k−1 u2k−2 uk+5 uk+4 uk+3 uk+2

Figure 13: Circular ladder CL2𝑘+1.

3.10. Betweenness Centrality of Vertices in Trees. In a tree, 𝐶𝐵 (V3)=C (5, 1) =5, there is exactly one path between any two vertices. Therefore 𝐶 (V )=C (1, 1, 4) =9. the betweenness centrality of a vertex is the number of paths 𝐵 5 passing through that vertex. A branch at a vertex V of a tree (32) 𝑇 is a maximal subtree containing V as an end vertex. The number of branches at V is deg(V). Example 13. Table 2 gives the possible values for the betweenness centrality of a vertex in a tree of 9 vertices. Theorem 11. The betweenness centrality 𝐶𝐵(V) of a vertex V in atree𝑇 is given by We consider the different possible combinations of the arguments in C so that the sum of arguments is 8. C (𝑛1,𝑛2,...,𝑛𝑘)=∑𝑛𝑖𝑛𝑗, 𝑖<𝑗 (31) 3.11. Betweenness Centrality of Vertices in Hypercubes. The 𝑛- 𝑛 or 𝑛-dimensional hypercube 𝑄𝑛 is defined recursively where the arguments 𝑖 denote the number of vertices of the 𝑛 V V by 𝑄1 =𝐾2 and 𝑄𝑛 =𝐾2 ×𝑄𝑛−1.Thatis,𝑄𝑛 =(𝐾2) branches at excluding , taken in any order. 𝑛 Harary [21]. 𝑄𝑛 is an 𝑛-regular graph containing 2 vertices 𝑛−1 Proof. Consider a vertex V in a tree 𝑇. Let there be 𝑘 and 𝑛2 edges. Each vertex can be labeled as a string of 𝑛 branches with number of vertices 𝑛1,𝑛2,...,𝑛𝑘 excluding V. bits 0 and 1.Twoverticesof𝑄𝑛 are adjacent if their binary The betweenness centrality of V in 𝑇 is the total number of representations differ at exactly one place (see Figure 15). The 𝑛 𝑛 paths passing through V. Since all the branches have only one 2 vertices are labeled by the 2 binary numbers from 0 to 𝑛 vertex V in common, excluding V, every path joining a pair of 2 −1. By definition, the length of a path between two vertices vertices of different branches passes through V.Thusthetotal 𝑢 and V isthenumberofedgesofthepath.Tomovefrom𝑢 to number of such pairs gives the betweenness centrality of V. V it suffices to cross successively the vertices whose labels are Hence C = ∑𝑖<𝑗 𝑛𝑖𝑛𝑗. thoseobtainedbychangingthebitsof𝑢 one by one in order to transform 𝑢 into V.If𝑢 and V differ only in 𝑖 bits, the distance Example 12. Consider the tree given in Figure 14. (hamming distance) between 𝑢 and V denoted by 𝑑(𝑢, V) is 𝑖 [22, 23]. For example, if 𝑢 = (101010) and V = (110011),then Here 𝑑(𝑢, V)=3. 𝑛 𝐶𝐵 (V1)=𝐶𝐵 (V4)=𝐶𝐵 (V6)=𝐶𝐵 (V7)=C (6) =0, There exists a path of length at most between any two vertices of 𝑄𝑛.Inotherwordsan𝑛-cube is a connected graph 𝐶𝐵 (V2)=C (1, 3, 2) = 11, of diameter 𝑛. The number of geodesics between 𝑢 and V is 10 International Journal of Combinatorics

6 7

5

1 2 3 4

Figure 14: A tree with 7 vertices.

110 100

10 010 0 00 000

101 111

11 001 011 1 01

Q0 Q1 Q2 Q3

0100 0110 1100 1110

0000 0010 1000 1010

0101 0111 1101 1111

0001 0011 1001 1011

Q4

Figure 15: Hypercubes.

given by the number of permutations 𝑑(𝑢, V)!.Ahypercube Theorem 14. The betweenness centrality of a vertex ina 𝑛−2 is bipartite and interval regular. For any two vertices 𝑢 and V, hypercube 𝑄𝑛 is given by 2 (𝑛 − 2) + 1/2. the interval 𝐼(𝑢, V) induces a hypercube of dimension 𝑑(𝑢, V) [24]. Another important property of 𝑛-cube is that it can be Proof. The hypercube 𝑄𝑛 of dimension 𝑛 is a vertex transitive 𝑛 2𝑛 constructed recursively from lower dimensional (see -regular graph containing vertices. Each vertex can be 𝑛 0 1 Figure 15). Consider two identical (𝑛−1)-cubes. Each (𝑛−1)- written as an tuple of binary digits and with adjacent 2𝑛−1 (𝑛−1) vertices differing in exactly one coordinate. The distance cube has vertices and each vertex has a labeling of - 𝑥 𝑦 (𝑥, 𝑦) bits. Join all identical pairs of vertices of the two cubes. Now between two vertices and denoted by is the number of places in which the corresponding coordinates of 𝑥 and 𝑦 increase the number of bits in the labels of all vertices by differ and the number of distinct geodesics between 𝑥 and 𝑦 placing 0 in the 𝑖th place of the first cube and 1 in the 𝑖th place 𝑛 is 𝑑(𝑥, 𝑦)! [22]. Let 0 = (0,0,...,0)be a vertex in 𝑄𝑛 whose of second cube. Thus we get an 𝑛-cube with 2 vertices, each betweenness centrality has to be determined. Consider all 𝑘- vertex having a label of 𝑛-bits and the corresponding vertices 0 2≤𝑘≤𝑛 𝑘 (𝑛−1) 𝑖 𝑛 subcubes containing the vertex for . Each - of the two -cubes differ only in the th bit. This -cube so subcube has vertices with 𝑛−𝑘zeros in their labels. Since 𝑛 (𝑛−1) constructedcanbeseenastheunionof pairs of -cubes each 𝑘-subcube can be distinguished by 𝑘 coordinates, the 𝑛 differing in exactly one position of bits. Thus the number of number of 𝑘-subcubes containing the vertex 0 is ( 𝑘 ).The (𝑛 − 1) 𝑛 2𝑛 -cubes in an -cube is .Theoperationofsplittingan vertex 0 lies on a geodesic joining a pair of vertices if and 𝑛-cube into two disjoint (𝑛 − 1)-cubes so that the vertices of onlyifthepairofverticesformsapairofantipodalvertices the two (𝑛−1)-cubes are in a one-to-one correspondence will of some subcube containing 0 [24]. So we consider all pairs be referred to as tearing [23]. Since there are 𝑛 bits, there are of antipodal vertices excluding the vertex 0 and its antipodal 𝑛 𝑛−𝑘 𝑛 directions for tearing. In general there are ( 𝑘 )2 number vertex in each 𝑘-subcube containing 0.Ifavertexofa𝑘- of 𝑘-subcubes associated with an 𝑛-cube. subcube has 𝑟 ones, then its antipodal vertex has 𝑘−𝑟ones. International Journal of Combinatorics 11

Table 2: Possible values for betweenness centrality in a tree of 9 The relative centrality and graph centrality are as follows: vertices. 󸀠 2𝐶𝐵 (V) Number of args. Possible combinations Values 𝐶 (V) = 𝐵 (2𝑛 −1)(2𝑛 −2) C(1,1,1,1,1,1,1,1) 8 28 (35) 7 C(2,1,1,1,1,1,1) 27 2𝑛−1 (𝑛−2) +1 = ,𝐶(𝐺) =0. C(2,2,1,1,1,1) 26 (2𝑛 −1)(2𝑛 −2) 𝐵 6 C(3,1,1,1,1,1) 25 C(2,2,2,1,1) 25 4. Conclusion 5 C(3,2,1,1,1) 24 C(4,1,1,1,1) 22 Betweenness centrality is a useful metric for analyzing graph C(2,2,2,2) 24 structures. When compared to other centrality measures, C(3,2,2,1) 23 computation of betweenness centrality is rather difficult as it involves calculation of the shortest paths between all 4 C(3,3,1,1) 22 C(5,1,1,1) pairs of vertices in a graph. We have derived expressions 18 for betweenness centrality of graphs which are the basic C(4,2,1,1) 21 components of larger and complex networks. This study is C(3, 3, 2) 21 therefore helpful for analysing larger classes of graphs. C(4, 2, 2) 20 C(4, 3, 1) 3 19 Conflict of Interests C(5, 2, 1) 17 C(6, 1, 1) 13 The authors declare that there is no conflict of interests C(4, 4) 16 regarding the publication of this paper. C(5, 3) 15 2 C(6, 2) 12 Acknowledgment C(7, 1) 7 1 C(8) 0 This work was supported by the University Grants Com- mission (UGC), Government of India, under the scheme of For any pair of such antipodal vertices there are 𝑘! geodesics Faculty Development Programme (FDP) for colleges. joining them and of that 𝑟!(𝑘 − 𝑟)! paths are passing through 𝑘 0.Thuseachpaircontributes𝑟!(𝑘 − 𝑟)!/𝑘!,thatis,1/ ( 𝑟 ),to References the betweenness centrality of 0. By symmetry, when 𝑘 is even the number of distinct pairs [1]E.OtteandR.Rousseau,“Socialnetworkanalysis:apowerful 𝑘 strategy, also for the information sciences,” Journal of Informa- of required antipodal vertices are given by ( 𝑟 ) for 1≤𝑟<𝑘/2 𝑘 tion Science,vol.28,no.6,pp.441–453,2002. and (1/2) ( 𝑟 ) for 𝑟=𝑘/2.When𝑘 is odd, the number of 𝑘 [2] K. Park and A. Yilmaz, “A social network an analysis approach distinct pairs of required antipodal vertices is given by ( 𝑟 ) for 1 ≤ 𝑟 ≤ (𝑘−1)/2. Taking all such pairs of antipodal vertices in to analyze road networks,” in Proceedings of the ASPRS Annual a 𝑘-subcube we get the contribution of betweenness centrality Conference, San Diego, Calif, USA, 2010. 𝑘/2−1 𝑘 𝑘 [3] V. Latora and M. Marchiori, “A measure of centrality based on as ∑ ( 𝑘 )(1/(𝑘 )) + (1/2) ( ) (1/ ( )) = (𝑘 − 1)/2, 𝑟=1 𝑟 𝑟 𝑘/2 𝑘/2 network efficiency,” New Journal of Physics,vol.9,article188, (𝑘−1)/2 𝑘 𝑘 when 𝑘 is even, and ∑𝑟=1 ( 𝑟 )(1/(𝑟 )) = (𝑘 − 1)/2,when 2007. 𝑘 is odd. Therefore considering all 𝑘-subcubes for 2≤𝑘≤𝑛, [4] E. Estrada, “Virtual identification of essential proteins within we get the betweenness centrality of 0 as the protein interaction network of yeast,” Proteomics,vol.6,no. 1,pp.35–40,2006. 𝑛 𝑘−1 𝑛 𝐶 (0) = ∑ ( )( ) [5] L. C. Freeman, “Centrality in social networks conceptual clari- 𝐵 2 𝑘 𝑘=2 fication,” Social Networks,vol.1,no.3,pp.215–239,1979. [6] D. Koschutzki¨ and F. Schreiber, “Centrality analysis methods 1 𝑛 𝑛 𝑛 𝑛 = [∑𝑘( )−∑ ( )] for biological networks and their application to gene regulatory 2 𝑘 𝑘 networks,” Gene Regulation and Systems Biology,vol.2008,no. 𝑘=2 𝑘=2 2, pp. 193–201, 2008. 1 =2𝑛−2 (𝑛−2) + . [7]A.M.M.Gonzalez,´ B. Dalsgaard, and J. M. Olesen, “Centrality 2 measures and the importance of generalist species in pollination (33) networks,” Ecological Complexity,vol.7,no.1,pp.36–43,2010. [8] M. E. J. Newman, “Scientific collaboration networks. II. sho rtest Therefore, for any vertex V, paths, weighted networks, and centrality,” Physical Review E, 1 vol.64,no.1,pp.1–7,2001. 𝐶 (V) =2𝑛−2 (𝑛−2) + . 𝐵 2 (34) [9]M.RubinovandO.Sporns,“Complexnetworkmeasuresof brain connectivity: uses and interpretations,” NeuroImage,vol. 52,no.3,pp.1059–1069,2010. 12 International Journal of Combinatorics

[10] P. Bonacich, “Power and centrality: a family of measures,” American Journal of Sociology,vol.92,no.5,pp.1170–1182,1987. [11] S. P. Borgatti, “Centrality and network flow,” Social Networks, vol.27,no.1,pp.55–71,2005. [12]S.P.BorgattiandM.G.Everett,“Agraph-theoreticperspective on centrality,” Social Networks,vol.28,no.4,pp.466–484,2006. [13] E. Estrada, D. J. Higham, and N. Hatano, “Communicabil- ity betweenness in complex networks,” Physica A: Statistical MechanicsanditsApplications, vol. 388, no. 5, pp. 764–774, 2009. [14] S. Gago, J. Coronicovˇ aHurajov´ a,´ and T. Madaras, “On betweenness-uniform graphs,” Czechoslovak Mathematical Journal,vol.63,no.3,pp.629–642,2013. [15] A. Bavelas, “Amathematical model for group structure. Human organization,” Applied Anthropology,vol.7,no.3,pp.16–30, 1948. [16] L. C. Freeman, “A set of measures of centrality based on betweenness,” Sociometry, vol. 40, pp. 35–41, 1977. [17] D. Cvetkovic, M. Doob, and S. Simic, “Generalized line graphs,” Journal of ,vol.5,no.4,pp.385–399,1981. [18] N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, UK, 1993. [19] H. Hosoya and F. Harary, “On the matching properties of three fence graphs,” Journal of Mathematical Chemistry,vol.12,no.1, pp. 211–218, 1993. [20] M. Noy and A. Ribo,´ “Recursively constructible families of graphs,” Advances in Applied Mathematics,vol.32,no.1-2,pp. 350–363, 2004. [21] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, USA, 1969. [22] S. Foldes, “A characterization of hypercubes,” Discrete Mathe- matics,vol.17,no.2,pp.155–159,1977. [23] Y. Saad and M. H. Schultz, “Topological properties of hyper- cubes,” IEEE Transactions on Computers,vol.37,no.7,pp.867– 872, 1988. [24] H. M. Mulder, “Interval-regular graphs,” Discrete Mathematics, vol. 41, no. 3, pp. 253–269, 1982. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

The Scientific International Journal of World Journal Differential Equations Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Submit your manuscripts at http://www.hindawi.com

International Journal of Advances in Combinatorics Mathematical Physics Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal of Journal of Mathematical Problems Abstract and Discrete Dynamics in Complex Analysis Mathematics in Engineering Applied Analysis Nature and Society Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

International Journal of Journal of Mathematics and Mathematical Discrete Mathematics Sciences

Journal of International Journal of Journal of Function Spaces Stochastic Analysis Optimization Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014