Lower Bounds of Forwarding Indices of Graph Products

A routing R of a connected graph G = (V,E) of order n consists in a set of n(n− 1) elementary paths R(u, v) specified for all ordered pairs u, v of vertices of G. To measure the efficiency of a routing deterministically, Chung, Coffman, Reiman and Simon [3] introduced the concept of forwarding index of a routing. The load of a vertex v (resp. an edge e) in a given routing R of G = (V,E), denoted by ξ(G,R, v) (resp. π(G,R, e)), is the number of paths of R going through v (resp. e), where v is not an end vertex. The parameters


Introduction
A routing R of a connected graph G = (V, E) of order n consists in a set of n(n − 1) elementary paths R(u, v) specified for all ordered pairs u, v of vertices of G.
To measure the efficiency of a routing deterministically, Chung, Coffman, Reiman and Simon [3] introduced the concept of forwarding index of a routing.
The load of a vertex v (resp.an edge e) in a given routing R of G = (V, E), denoted by ξ(G, R, v) (resp.π(G, R, e)), is the number of paths of R going through v (resp.e), where v is not an end vertex.The parameters are defined as the vertex forwarding (resp.the edge forwarding) index of G with respect to R, and are defined as the vertex forwarding (resp.the edge forwarding) index of G.The original research of the forwarding indices is motivated by the fact that maximizing network capacity can be reduced to minimizing vertex-forwarding index or edge-forwarding index of a routing.Thus, the forwarding index problem has been studied widely by many researchers (see, for example, [1,12]).
Although, determining the forwarding index problem has been shown to be N P -complete by Saad [11].The exact values of the forwarding index of many important classes of graphs have been computed [2,4,5,6,8].For more complete results on forwarding indices, we can refer to the survey of Xu et Al.[15].
For a given connected graph G = (V, E) of order n and number of edges m, set where d G (u, v) denotes the distance from the vertex u to the vertex v in G. Let d(G, k) be the number of pairs of vertices of a graph G that are at distance k.
The paper is organized as follows, in section 2, we present the graph products.Our main results on graph products are presented in section 3, we compute lower bounds of forwarding indices for join, composition, disjunction and symmetric difference of graphs.In section 4, lower bounds of several operators on connected graphs, such as subdivision graph and total graphs are computed.

Graph Products
The Cartesian product G × H of graphs G and H is a graph with vertex set V (G) × V (H), and any two vertices (a, b) and (u, v) are adjacent in G × H if and only if either a = u and b is adjacent with v, or b = v and a is adjacent with u, see [10] for details.
The join G = G + H of graphs G and H with disjoint vertex sets V (G) and V (H) and edge sets E(G) and E(H) is the graph union G ∪ H together with all the edges joining V (G) and V (H).
The composition G[H] of graphs G and H with disjoint vertex sets V (G) and V (H) and edge sets E(G) and E(H) is the graph with vertex set V (G) × V (H) and u = (u 1 , v 1 ) is adjacent with v = (u 2 , v 2 ) whenever (u 1 is adjacent with u 2 ) or (u 1 = u 2 and v 1 is adjacent with v 2 ), [10].

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The disjunction G ∨ H of graphs G and H is the graph with vertex set The symmetric difference G ⊕ H of two graphs G and H is the graph with vertex set The following lemma is related to distance properties of some graph products.
Lemma 2.1.(Hossein-Zadeh et Al.[9]) Let G and H be graphs.Then we have: 4. The Cartesian product, join, composition, disjunction and symmetric difference of graphs are associative and all of them are commutative except from composition. 5.

Main results
In this section, some exact formulae for expressions A(G) and B(G) of the Cartesian product, composition, join, disjunction and symmetric difference of graphs are presented.
The Forwarding indices of the Cartesian product graphs were studied in [14].In the following propositions, we compute the lower bounds of the forwarding indices A(G) and B(G) for known product graphs.Theorem 3.1.Let G and H be connected graphs.Then Proof.By Lemma 2.1, we have: and hence the result.
Further, we have that proving the result.
Proof.By Lemma 2.1, we have: Proof.By Lemma 2.1, we have: The graphs S(G) and T (G) are called the subdivision and total graph of G, respectively.For more details on these operations we refer the reader to [13,16].
In this section, we compute lower bounds of forwarding indices A(G) and B(G) for some graph operations.We use the following lemma in [16].Lemma 4.2.(Yan et Al 2007 [16]) Let G be a graph, then
Theorem 3.4.Let G and H be connected graphs.Then A