- Email: [email protected]
Betweenness Centrality in Cartesian Product of Graphs
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Electronic Notes in Discrete Mathematics 63 (2017) 287–294 www.elsevier.com/locate/endm
Betweenness Centrality in Cartesian Product of Graphs Sunil Kumar R 1,2 Department of Computer Applications Cochin University of Science and Technology Cochin, India
Kannan Balakrishnan 3 Department of Computer Applications Cochin University of Science and Technology Cochin, India
Abstract Betweenness centrality is a widely used to measure the potential of a node to control the communication over the network. It measures the extent to which a vertex is part of the shortest paths between pairs of other vertices in a graph. In this paper, we establish expression for betweenness centrality in Cartesian product of graphs. And investigate the same on certain graphs such as Hamming graphs, hypercubes, Cartesian product of path and cycle. Keywords: Betweenness centrality, Pairwise dependency, Cartesian product, Geodetic graph.
1
The work of the author is supported by the University Grants Commission (UGC), Government of India, under the scheme of Faculty Development Programme (FDP) for colleges. 2 Email: [email protected] 3 Email: [email protected] https://doi.org/10.1016/j.endm.2017.11.024 1571-0653/© 2017 Elsevier B.V. All rights reserved.
288
1
S. Kumar R., K. Balakrishnan / Electronic Notes in Discrete Mathematics 63 (2017) 287–294
Introduction
Several Centrality measures have so far been studied and their importance is increasing day by day. Betweenness centrality has a vital role in the analysis of networks [4, 9–11] Betweennes centrality [1, 2] indicates the betweenness of a vertex (or an edge) in a network. The computation of this index based on enumeration as per the definition becomes impractical as the number of vertices n increases. Its complexity is of order O(n3 ). But a large network can be thought of as it is made by joining smaller networks together. There are several graph operations which results in a larger graph G and many of the properties of larger graphs can be derived from their constituent graphs. Cartesian product is an important graph operation. It is assumed that the graphs under consideration are simple undirected connected graphs. A definition to betweenness centrality of a vertex in a graph G, given by Freeman [5] is as follows Definition 1.1 [Betweenness Centrality] If x ∈ V (G), the betweenness centrality B(x) for x is defined as B(x) = δ(u, v|x) {u,v}⊆V (G) u=v=x
where δ(u, v|x) is called the pair-dependency of {u, v} on x and it is defined where σ(u, v) is the number of shortest u-v paths and as δ(u, v|x) = σ(u,v|x) σ(u,v) σ(u, v|x) is the number of shortest u-v paths containing x. Observe that x ∈ V (G) lies on the shortest u-v path iff d(u, v) = d(u, x) + d(x, v). The number of shortest u-v paths passing through x is given by σ(u, v|x) = σ(u, x) × σ(x, v)
Definition 1.2 [Cartesian Product] [13] The Cartesian product of two graphs G and H, denoted by GH, is a graph with vertex set V (G) × V (H), where two vertices (g, h) and (g , h ) are adjacent if g = g and hh ∈ E(H), or gg ∈ E(G) and h = h . The graphs G and H are called factors of the product GH. For any h ∈ V (H), the subgraph of GH induced by V (G) × {h} is called G-fiber or G-layer, denoted by Gh . Similarly, we can define H-fiber or
S. Kumar R., K. Balakrishnan / Electronic Notes in Discrete Mathematics 63 (2017) 287–294
289
H-layer. They are isomorphic to G and H respectively. GH contains |H| copies of G and |G| copies of H. Projections are the maps from a product graph to its factors. They are weak homomorphisms in the sense that they respect adjacency. The two projections on GH namely pG : GH → G and pH : GH → H defined by pG (g, h) = g and pH (g, h) = h refer to the corresponding G-, H- coordinates. Thus an edge in GH is mapped into a single vertex, by one of the projections pG or pH and into an edge by the other. If G and H are connected, then GH is also connected. Assuming isomorphic graphs are equal, Cartesian product is commutative as well as associative. The following proposition shows that the distance between two vertices in the product graph is the sum of the distance between their projections in the factor graphs. Lemma 1.3 (Distance lemma) [7] If (g, h) and (g , h ) are vertices of a Cartesian product GH, then dGH (g, h), (g , h ) = dG (g, g ) + dH (h, h ) (1) Proposition 1.4 [8] Let v1 = (g1 , h1 ) and v2 = (g2 , h2 ) be two vertices of GH, then the vertex v3 = (g3 , h3 ) lies in IGH (v1 , v2 ) if and only if g3 ∈ IG (g1 , g2 ) and h3 ∈ IH (h1 , h2 ). Definition 1.5 A graph G is a geodetic graph [12] if every pair of vertices of G is connected by a unique shortest path.
2
Betweenness centrality of vertices in Cartesian product of graphs
The following proposition shows how the number of geodesics between two vertices u and v in a product graph GH is related to the the number of geodesics between their projections in the factor graphs. Proposition 2.1 If u = (g, h) and v = (g , h ) are vertices in GH, then the number of shortest u-v paths in GH is given by dG (g, g ) + dH (h, h ) σGH (g, h), (g , h ) = σG (g, g ) × σH (h, h ) × (2) dG (g, g ) Proof. Consider the vertices u = (g, h) and v = (g , h ) in GH. Let d = d(u, v) denotes the distance between u and v in GH. Suppose there exists unique shortest paths between g and g in G and h and h in H. A shortest path from u to v is a sequence of d edges and the image of each edge under
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the projections pG and pH is an edge lying between g and g or h and h . If the sequence of d edges of the u-v path in GH makes a sequence of dG edges in G and a sequence of dH edges in H, then d = dG + dH . Since dG and dH are the same for any shortest u-v path, the number of shortest paths between u and v in GH is the number of ways of selecting dG edges (or dH edges) from d edges which is ddG . If there exists σG shortest paths between g and g in G and σH shortest between h and h in H, then corresponding to dpaths each pair, there exists dG shortest paths between u and v in GH. H (h,h ) Therefore, σGH (g, h), (g , h ) = σG (g, g ) × σH (h, h ) × dG (g,gdG)+d 2 (g,g ) For brevity, we may write σGH = σG × σH ×
d
GH
dG
Corollary 2.2 If G and H are geodetic graphs, then the number of shortest paths between (g, h) and (g , h ) in GH is given by dG (g, g ) + dH (h, h ) σGH (g, h), (g , h ) = dG (g, g ) By the associativity of , Equation 2 can be generalized as Proposition 2.3 Let G = ki=1 Gni . If u = (u1 , u2 , . . . , uk ), v = (v1 , v2 , . . . , vk ) are two vertices in G such that σGi (ui , vi ) = σi , dGi (ui , vi ) = di and d = di , then d d − d1 d − d1 − d2 ...1 σG (u, v) = σ1 σ2 . . . σn d1 d2 d3 Corollary 2.4 Let G = ki=1 Gni where each Gni are geodetic. If u = (u1 , u2 , . . . , uk ), v = (v1 , v2 , . . . , vk ) are two vertices in G such that dGi (ui , vi ) = di and d= di , then d − d1 d − d1 − d2 d σG (u, v) = ...1 d1 d2 d3 Proposition 2.5 Let v1 ,v2 and v3 are any three vertices in GH. Then σGH (v1 , v2 |v3 ) = σGH (v1 , v3 ) × σGH (v3 , v2 )
(3)
Theorem 2.6 The betweenness centrality of x = (g0 , h0 ) in GH is given by BGH (x) =
u=v=x {u,v}⊆V (GH)
δGH (u, v|x)
S. Kumar R., K. Balakrishnan / Electronic Notes in Discrete Mathematics 63 (2017) 287–294
where δGH (u, v|x) = δG (g, g |g0 ) × δH (h, h |h0 )) ×
291
D1 × D2 D
with dG (g, g0 ) + dH (h, h0 ) dG (g0 , g ) + dH (h0 , h ) , D2 = dG (g, g0 ) dG (g0 , g )
D1 = ,
D=
dG (g, g ) + dH (h, h ) dG (g, g )
Proof. The result follows from the definition of betweenness centrality and from Equations [1 - 3] 2 Definition 2.7 [Wiener index of a graph] [14] The Wiener index of a graph G, denoted by W (G) is the sum of the distances between all (unordered) pairs of vertices of G. i.e., W (G) = d(vi , vj ) i
Wiener index is also known as total status or total distance of a graph. The Wiener index of Cartesian product [3, 6, 15] of two graphs G and H is given by W (GH) = |G|2 W (H) + |H|2 W (G) It can be extended to W (ni=1 Gi )
=
n
i=1
j=i
W (Gi )
|Gj |2
The betweenness centrality of G is given by v∈V (G)
|G| B(v) = W (G) − 2
If G is vertex transitive, the betweenness centrality of v ∈ G is given by W (G) − B(v) = |G|
|G| 2
292
2.1
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Hamming graphs
Hamming graphs are Cartesian products of complete graphs.i.e, H = ri=1 Kni for some r ≥ 1 and ni ≥ 2. The vertices of H can be labeled with vector (a1 , a2 , . . . ar ) where ai ∈ {0, 1, . . . , ni − 1}. Two vertices of H are adjacent if the corresponding vectors differ in precisely one coordinate. The distance (named Hamming distance) between two vertices u and v denoted by d(u, v) is the number of positions in which the two vectors differ. Hypercubes are Cartesian product of complete graphs K2 . Proposition 2.8 Let H be the Hamming graph given by H = ri=1 Kni , then for v ∈ H r r 1 1 1 B(v) = + ni r − 1 − 2 i=1 ni 2 i=1 Proposition 2.9 If G is the Cartesian product of r even cycles. i.e, G = ri=1 Cni , ni ∈ 2Z+ , then for v ∈ G r r r
1 B(v) = ni ni − 4 ni − 1 8 i=1 i=1 i=1
if ni = 2ki , B(v) = 2r−2
r i=1
ki
r
i=1
1 ki − 2 + 2
Proposition 2.10 If G is the Cartesian product of r odd cycles. i.e, G = ri=1 Cni , ni ∈ 2Z+ + 1, then for v ∈ G B(v) =
2.2
r r r
1
1 ni − 1 ni − −4 ni 8 i=1 ni i=1 i=1
Product of path and cycle
Theorem 2.11 Let G = Pm Cn where V (Pm ) = {0, 1, . . . , m − 1} and V (Cn ) = {0, 1, . . . , n − 1}. Then for (i, j) ∈ G Case 1 If (i, j) ∈ Pm C2k+1 i B[(i, j)] = k2 +i(1+2k)(m−i−1)+2 k+1 (H −H )+H −H m−i+r 1+r 1+i 1 r=0 2 Case 2 If (i, j) ∈ Pm C2k
S. Kumar R., K. Balakrishnan / Electronic Notes in Discrete Mathematics 63 (2017) 287–294
B[(i, j)] =
3
(k−1)2 2
+ 2ik(m − i − 1) + k 2
293
i r=0 (Hm−i+r − H1+r ) + H1+i − H1
Conclusion
A composite graph can be constructed by applying different graph operations on smaller graphs. Many of the structural properties of the composite graphs can be studied from its constituent smaller graphs. An expression for the betweenness centrality of Cartesian product of graphs has been derived and examined for different graph products. This may be extended to other graphproducts.
References [1] Borgatti, S. P., and M. G. Everett, A graph-theoretic perspective on centrality, Social networks, 28(2006), 466–484. [2] Brandes, U., On variants of shortest-path betweenness centrality and their generic computation, Social Networks, 30(2008), 136–145. [3] Dehmer, M., and F. Emmert-Streib “Quantitative Graph Theory: Mathematical Foundations and Applications,” CRC Press, 2014 [4] Estrada, E., Virtual identification of essential proteins within the protein interaction network of yeast, Proteomics, 6(2006), 35–40 [5] Freeman, L. C., A set of measures of centrality based on betweenness, Sociometry, (1977), 35–41. [6] Graovac, A., and T. Pisanski, On the wiener index of a graph, Journal of mathematical chemistry, 8(1991), 53–62. [7] Hammack, R., W. Imrich and S. Klavzar, “Handbook of product graphs”, CRC Press, 2011. [8] Imrich, W., S. Klavzar and D. F. Rall, “Topics in graph theory: Graphs and their Cartesian product”, CRC Press, 2008. [9] Jord an, F., Keystone species and food webs, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 364(2009), 1733–1741. [10] Krebs, V. E., Mapping networks of terrorist cells, Connections, 24(2002), 43–52. [11] Liljeros, F., C. R. Edling, L. A. N. Amaral, H. E. Stanley and Y. Aberg, The web of human sexual contacts, Nature, 411(2001), 907–908.
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[12] Ore, O., Theory of graphs, American Mathematical Society Colloquium Publications, 38(1962). [13] Sabidussi, G., Graphs with given group and given graph-theoretical properties, Canad. J. Math, 9(1957), C525. [14] Wiener, H., Structural determination of paraffin boiling points, Journal of the American Chemical Society, 69(1947), 17–20. [15] Yeh, Y.-N; and I. Gutman, On the sum of all distances in composite graphs, Discrete Mathematics, 135(1994), 359–365.
Electronic Notes in Discrete Mathematics 63 (2017) 287–294 www.elsevier.com/locate/endm
Betweenness Centrality in Cartesian Product of Graphs Sunil Kumar R 1,2 Department of Computer Applications Cochin University of Science and Technology Cochin, India
Kannan Balakrishnan 3 Department of Computer Applications Cochin University of Science and Technology Cochin, India
Abstract Betweenness centrality is a widely used to measure the potential of a node to control the communication over the network. It measures the extent to which a vertex is part of the shortest paths between pairs of other vertices in a graph. In this paper, we establish expression for betweenness centrality in Cartesian product of graphs. And investigate the same on certain graphs such as Hamming graphs, hypercubes, Cartesian product of path and cycle. Keywords: Betweenness centrality, Pairwise dependency, Cartesian product, Geodetic graph.
1
The work of the author is supported by the University Grants Commission (UGC), Government of India, under the scheme of Faculty Development Programme (FDP) for colleges. 2 Email: [email protected] 3 Email: [email protected] https://doi.org/10.1016/j.endm.2017.11.024 1571-0653/© 2017 Elsevier B.V. All rights reserved.
288
1
S. Kumar R., K. Balakrishnan / Electronic Notes in Discrete Mathematics 63 (2017) 287–294
Introduction
Several Centrality measures have so far been studied and their importance is increasing day by day. Betweenness centrality has a vital role in the analysis of networks [4, 9–11] Betweennes centrality [1, 2] indicates the betweenness of a vertex (or an edge) in a network. The computation of this index based on enumeration as per the definition becomes impractical as the number of vertices n increases. Its complexity is of order O(n3 ). But a large network can be thought of as it is made by joining smaller networks together. There are several graph operations which results in a larger graph G and many of the properties of larger graphs can be derived from their constituent graphs. Cartesian product is an important graph operation. It is assumed that the graphs under consideration are simple undirected connected graphs. A definition to betweenness centrality of a vertex in a graph G, given by Freeman [5] is as follows Definition 1.1 [Betweenness Centrality] If x ∈ V (G), the betweenness centrality B(x) for x is defined as B(x) = δ(u, v|x) {u,v}⊆V (G) u=v=x
where δ(u, v|x) is called the pair-dependency of {u, v} on x and it is defined where σ(u, v) is the number of shortest u-v paths and as δ(u, v|x) = σ(u,v|x) σ(u,v) σ(u, v|x) is the number of shortest u-v paths containing x. Observe that x ∈ V (G) lies on the shortest u-v path iff d(u, v) = d(u, x) + d(x, v). The number of shortest u-v paths passing through x is given by σ(u, v|x) = σ(u, x) × σ(x, v)
Definition 1.2 [Cartesian Product] [13] The Cartesian product of two graphs G and H, denoted by GH, is a graph with vertex set V (G) × V (H), where two vertices (g, h) and (g , h ) are adjacent if g = g and hh ∈ E(H), or gg ∈ E(G) and h = h . The graphs G and H are called factors of the product GH. For any h ∈ V (H), the subgraph of GH induced by V (G) × {h} is called G-fiber or G-layer, denoted by Gh . Similarly, we can define H-fiber or
S. Kumar R., K. Balakrishnan / Electronic Notes in Discrete Mathematics 63 (2017) 287–294
289
H-layer. They are isomorphic to G and H respectively. GH contains |H| copies of G and |G| copies of H. Projections are the maps from a product graph to its factors. They are weak homomorphisms in the sense that they respect adjacency. The two projections on GH namely pG : GH → G and pH : GH → H defined by pG (g, h) = g and pH (g, h) = h refer to the corresponding G-, H- coordinates. Thus an edge in GH is mapped into a single vertex, by one of the projections pG or pH and into an edge by the other. If G and H are connected, then GH is also connected. Assuming isomorphic graphs are equal, Cartesian product is commutative as well as associative. The following proposition shows that the distance between two vertices in the product graph is the sum of the distance between their projections in the factor graphs. Lemma 1.3 (Distance lemma) [7] If (g, h) and (g , h ) are vertices of a Cartesian product GH, then dGH (g, h), (g , h ) = dG (g, g ) + dH (h, h ) (1) Proposition 1.4 [8] Let v1 = (g1 , h1 ) and v2 = (g2 , h2 ) be two vertices of GH, then the vertex v3 = (g3 , h3 ) lies in IGH (v1 , v2 ) if and only if g3 ∈ IG (g1 , g2 ) and h3 ∈ IH (h1 , h2 ). Definition 1.5 A graph G is a geodetic graph [12] if every pair of vertices of G is connected by a unique shortest path.
2
Betweenness centrality of vertices in Cartesian product of graphs
The following proposition shows how the number of geodesics between two vertices u and v in a product graph GH is related to the the number of geodesics between their projections in the factor graphs. Proposition 2.1 If u = (g, h) and v = (g , h ) are vertices in GH, then the number of shortest u-v paths in GH is given by dG (g, g ) + dH (h, h ) σGH (g, h), (g , h ) = σG (g, g ) × σH (h, h ) × (2) dG (g, g ) Proof. Consider the vertices u = (g, h) and v = (g , h ) in GH. Let d = d(u, v) denotes the distance between u and v in GH. Suppose there exists unique shortest paths between g and g in G and h and h in H. A shortest path from u to v is a sequence of d edges and the image of each edge under
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the projections pG and pH is an edge lying between g and g or h and h . If the sequence of d edges of the u-v path in GH makes a sequence of dG edges in G and a sequence of dH edges in H, then d = dG + dH . Since dG and dH are the same for any shortest u-v path, the number of shortest paths between u and v in GH is the number of ways of selecting dG edges (or dH edges) from d edges which is ddG . If there exists σG shortest paths between g and g in G and σH shortest between h and h in H, then corresponding to dpaths each pair, there exists dG shortest paths between u and v in GH. H (h,h ) Therefore, σGH (g, h), (g , h ) = σG (g, g ) × σH (h, h ) × dG (g,gdG)+d 2 (g,g ) For brevity, we may write σGH = σG × σH ×
d
GH
dG
Corollary 2.2 If G and H are geodetic graphs, then the number of shortest paths between (g, h) and (g , h ) in GH is given by dG (g, g ) + dH (h, h ) σGH (g, h), (g , h ) = dG (g, g ) By the associativity of , Equation 2 can be generalized as Proposition 2.3 Let G = ki=1 Gni . If u = (u1 , u2 , . . . , uk ), v = (v1 , v2 , . . . , vk ) are two vertices in G such that σGi (ui , vi ) = σi , dGi (ui , vi ) = di and d = di , then d d − d1 d − d1 − d2 ...1 σG (u, v) = σ1 σ2 . . . σn d1 d2 d3 Corollary 2.4 Let G = ki=1 Gni where each Gni are geodetic. If u = (u1 , u2 , . . . , uk ), v = (v1 , v2 , . . . , vk ) are two vertices in G such that dGi (ui , vi ) = di and d= di , then d − d1 d − d1 − d2 d σG (u, v) = ...1 d1 d2 d3 Proposition 2.5 Let v1 ,v2 and v3 are any three vertices in GH. Then σGH (v1 , v2 |v3 ) = σGH (v1 , v3 ) × σGH (v3 , v2 )
(3)
Theorem 2.6 The betweenness centrality of x = (g0 , h0 ) in GH is given by BGH (x) =
u=v=x {u,v}⊆V (GH)
δGH (u, v|x)
S. Kumar R., K. Balakrishnan / Electronic Notes in Discrete Mathematics 63 (2017) 287–294
where δGH (u, v|x) = δG (g, g |g0 ) × δH (h, h |h0 )) ×
291
D1 × D2 D
with dG (g, g0 ) + dH (h, h0 ) dG (g0 , g ) + dH (h0 , h ) , D2 = dG (g, g0 ) dG (g0 , g )
D1 = ,
D=
dG (g, g ) + dH (h, h ) dG (g, g )
Proof. The result follows from the definition of betweenness centrality and from Equations [1 - 3] 2 Definition 2.7 [Wiener index of a graph] [14] The Wiener index of a graph G, denoted by W (G) is the sum of the distances between all (unordered) pairs of vertices of G. i.e., W (G) = d(vi , vj ) i
Wiener index is also known as total status or total distance of a graph. The Wiener index of Cartesian product [3, 6, 15] of two graphs G and H is given by W (GH) = |G|2 W (H) + |H|2 W (G) It can be extended to W (ni=1 Gi )
=
n
i=1
j=i
W (Gi )
|Gj |2
The betweenness centrality of G is given by v∈V (G)
|G| B(v) = W (G) − 2
If G is vertex transitive, the betweenness centrality of v ∈ G is given by W (G) − B(v) = |G|
|G| 2
292
2.1
S. Kumar R., K. Balakrishnan / Electronic Notes in Discrete Mathematics 63 (2017) 287–294
Hamming graphs
Hamming graphs are Cartesian products of complete graphs.i.e, H = ri=1 Kni for some r ≥ 1 and ni ≥ 2. The vertices of H can be labeled with vector (a1 , a2 , . . . ar ) where ai ∈ {0, 1, . . . , ni − 1}. Two vertices of H are adjacent if the corresponding vectors differ in precisely one coordinate. The distance (named Hamming distance) between two vertices u and v denoted by d(u, v) is the number of positions in which the two vectors differ. Hypercubes are Cartesian product of complete graphs K2 . Proposition 2.8 Let H be the Hamming graph given by H = ri=1 Kni , then for v ∈ H r r 1 1 1 B(v) = + ni r − 1 − 2 i=1 ni 2 i=1 Proposition 2.9 If G is the Cartesian product of r even cycles. i.e, G = ri=1 Cni , ni ∈ 2Z+ , then for v ∈ G r r r
1 B(v) = ni ni − 4 ni − 1 8 i=1 i=1 i=1
if ni = 2ki , B(v) = 2r−2
r i=1
ki
r
i=1
1 ki − 2 + 2
Proposition 2.10 If G is the Cartesian product of r odd cycles. i.e, G = ri=1 Cni , ni ∈ 2Z+ + 1, then for v ∈ G B(v) =
2.2
r r r
1
1 ni − 1 ni − −4 ni 8 i=1 ni i=1 i=1
Product of path and cycle
Theorem 2.11 Let G = Pm Cn where V (Pm ) = {0, 1, . . . , m − 1} and V (Cn ) = {0, 1, . . . , n − 1}. Then for (i, j) ∈ G Case 1 If (i, j) ∈ Pm C2k+1 i B[(i, j)] = k2 +i(1+2k)(m−i−1)+2 k+1 (H −H )+H −H m−i+r 1+r 1+i 1 r=0 2 Case 2 If (i, j) ∈ Pm C2k
S. Kumar R., K. Balakrishnan / Electronic Notes in Discrete Mathematics 63 (2017) 287–294
B[(i, j)] =
3
(k−1)2 2
+ 2ik(m − i − 1) + k 2
293
i r=0 (Hm−i+r − H1+r ) + H1+i − H1
Conclusion
A composite graph can be constructed by applying different graph operations on smaller graphs. Many of the structural properties of the composite graphs can be studied from its constituent smaller graphs. An expression for the betweenness centrality of Cartesian product of graphs has been derived and examined for different graph products. This may be extended to other graphproducts.
References [1] Borgatti, S. P., and M. G. Everett, A graph-theoretic perspective on centrality, Social networks, 28(2006), 466–484. [2] Brandes, U., On variants of shortest-path betweenness centrality and their generic computation, Social Networks, 30(2008), 136–145. [3] Dehmer, M., and F. Emmert-Streib “Quantitative Graph Theory: Mathematical Foundations and Applications,” CRC Press, 2014 [4] Estrada, E., Virtual identification of essential proteins within the protein interaction network of yeast, Proteomics, 6(2006), 35–40 [5] Freeman, L. C., A set of measures of centrality based on betweenness, Sociometry, (1977), 35–41. [6] Graovac, A., and T. Pisanski, On the wiener index of a graph, Journal of mathematical chemistry, 8(1991), 53–62. [7] Hammack, R., W. Imrich and S. Klavzar, “Handbook of product graphs”, CRC Press, 2011. [8] Imrich, W., S. Klavzar and D. F. Rall, “Topics in graph theory: Graphs and their Cartesian product”, CRC Press, 2008. [9] Jord an, F., Keystone species and food webs, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 364(2009), 1733–1741. [10] Krebs, V. E., Mapping networks of terrorist cells, Connections, 24(2002), 43–52. [11] Liljeros, F., C. R. Edling, L. A. N. Amaral, H. E. Stanley and Y. Aberg, The web of human sexual contacts, Nature, 411(2001), 907–908.
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[12] Ore, O., Theory of graphs, American Mathematical Society Colloquium Publications, 38(1962). [13] Sabidussi, G., Graphs with given group and given graph-theoretical properties, Canad. J. Math, 9(1957), C525. [14] Wiener, H., Structural determination of paraffin boiling points, Journal of the American Chemical Society, 69(1947), 17–20. [15] Yeh, Y.-N; and I. Gutman, On the sum of all distances in composite graphs, Discrete Mathematics, 135(1994), 359–365.