Distance-Regular Graphs and Distance Based Graph

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edge of G, and any two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint in G. One can also define iterated line ...
The First Conference on Computational Group Theory, Computational Number Theory and Applications, University of Kashan, 26-28 Azar, 1393 (December 17-19 2014), pp: 151-154.

Oral Presentation

Distance-Regular Graphs and Distance Based Graph Invariants Reza Sharafdini Department of Mathematics, Faculty of Sciences, Persian Gulf University, Bushehr, Iran [email protected]

Abstract In this article we aim to obtain an explicit formula for some distance based graph invariants of distance-regular graphs. In fact we obtain formulas for Wiener index and its multiplicative version of a distance-regular graph in terms of its intersection array and its distance partition.

Keywords: Distance-regular, strongly regular, Wiener index, multiplicative Wiener index, distancebalanced, Szeged index. MSC(2010): Primary: 05C12, 05C31.

1

Introduction

Throughout this paper G = (V, E) denotes a connected, simple and finite graph with vertex set V = V (G) and edge set E = E(G). The distance d(u, v) between two vertices u and v is the minimum of the lengths of paths between u and v. The diameter D of a graph G is defined as D := maxu,v∈V (G) d(u, v). For a graph G of n o diameter D, vertex v ∈ V (G), and for 0 ≤ i ≤ D, define Gi (v) = w ∈ V | d(v, w) = i . These cells G0 (v), G1 (v), . . . , GD (v) form a distance partition (or a level decomposition) of G based on v ∈ G. For each v ∈ V (G), G1 (v) is called the set of neighbors of v; and the size of G1 (v) is called the degree of v. A graph is said to be k-regular if |G1 (u)| = |G1 (v)| = k for all u, v ∈ V (G). A distance-regular graph is a simple connected graph such that for any two vertices u and v, the number of vertices 151

at distance i from u and at distance j from v depends only upon i, j, and t = d(v, w). Equivalently, a distance-regular graph is a simple connected graph of diameter D for which there exist integers ai , bi , ci , i = 0, . . . , D such that for any two vertices x, y in V (G) at distance i = d(x, y), there are exactly ci neighbors of y in Gi−1 (x), bi neighbors of y in Gi+1 (x) and ai neighbors of y in Gi (x). Namely,   ai if j = i; ci if j = i − 1; | G1 (y) ∩ G j (x) |=  bi if j = i + 1.   c0 c1 · · · cD−1 cD  The numbers ai , bi , and ci are often displayed in a three-line array a0 a1 · · · aD−1 aD ,   b0 b1 · · · bD−1 bD which is known as its intersection array. In particular G is regular of degree k := b0 and a0 = c0 = bD = 0, , c1 = 1, ai + bi + ci = k 0 ≤ i ≤ D. We may represent the intersection arrays of a distance-regular graph as {b0 = k, b1 , ..., bD−1 ; c1 = 1, c2 , ..., cD }. Suppose that G is a distance-regular graph of diameter D with the intersection array {b0 = k, b1 , ..., bD−1 ; c1 = 1, c2 , ..., cD }. Fixing 0 ≤ i ≤ D, by the definition of distance-regular graphs, the size of Gi (u) does not depend on the choice of u ∈ V (G). Let us denote the size of Gi (u) by ki , i.e., ki := |Gi (u)|, 0 ≤ i ≤ D. Note that k0 = |G0 (u)| = 1 and k1 = |G1 (u)| = k and 1 + k + k2 + . . . + kD = |V (G)|.

(1.1)

Moreover, For any vertex u ∈ V (G), any vertex of Gi (u) is adjacent to bi vertices in Gi+1 (u) and any vertex of Gi+1 (u) is adjacent to ci vertices in Gi (u). Thus by two way of counting the number of edges between Gi (u) and Gi+1 (u) we have: ki bi =| Gi (u) | bi =| Gi+1 (u) | ci+1 = ki+1 ci+1 .

(1.2)

Hence, it follows from (1.2) that the number of vertices at distance i of a vertex u, namely |Gi (u)|, is obtained directly from the intersection array ([1, Proposition 20.4]) ki = |Gi (u)| =

∏i−1 j=0 b j ∏ij=2 c j

(2 ≤ i ≤ D) and |G1 (u)| = b0 .

(1.3)

The problem of distances in graph attracts the attention of scientist both as theory and applications. In 1947, H. Wiener [18] has proposed his path number, as the total distance between all carbon atoms for correlating with the thermodynamic properties of alkanes. Numerous of its chemical applications were reported and its mathematical properties are well understood. This index now is called the Wiener index W (G) of a graph G, and defined as the sum of distances between all unordered pairs of vertices of G, i.e., W (G) := ∑{u,v}⊆V d(u, v). In fact, if we denote by d(G, k), k ≥ 0, the number of unordered vertex pairs at distance k, then W (G) = ∑D k=1 k · d(G, k). Note that d(G, 3) is called the Wiener polarity of G which is some times denoted by Wp (G). For u ∈ V (G), the distance sum D(u) and its multiplication version D∗ (u) of u is defined as D(u) = ∑v∈V (G) d(u, v), D∗ (u) = ∏v∈V (G) d(u, v). In this case the Wiener index of G and its v6=u

multiplicative version are represented as follows: W (G) =

1 ∑ D(u), 2 u∈V (G) 152

(1.4)

W ∗ (G) =

1 ∏ D∗ (u). 2 u∈V (G)

(1.5)

The following modification of Wiener index has also been considered: Wλ (G) :=

d(u, v)λ ;



λ 6= 0.

{u,v}⊆V

Wλ (G) :=

1 ∑ Dλ (u); 2 {u,v}∈V (G)

λ 6= 0,

where Dλ (u) is called λ -distance sum of u and defined as follows: Dλ (u) = ∑{u,v}∈V (G) d(u, v)λ The multiplicative version of Wiener index of G, denoted by W ∗ (G) is also defined as follows [4]: W ∗ (G) := ∏v∈V (G) d(u, v) = ∏D k=1 k · d(G, k). Hosoya [5] introduced a distance-based graph polynov6=u

mial H(G, x) = ∑k≥1 d(G, k)xk , nowadays called the Hosoya polynomial. It is easy to check that it can be written in the following form H(G, x) = ∑{u,v}⊆V (G) xd(u,v) . The first derivative of the Hosoya polynomial at = 1 is equal to the Wiener index.

2

Main Results

Theorem 2.1. Let G be a distance-regular array is {b0 , b1 , ..., bD−1 ; c1 =  graph whose intersection  1, c2 , ..., cD }. Then we have Wλ (G) =

nb0 2

λ 1 + ∑D i=2 i

∏i−1 j=1 b j ∏ij=2 c j

.

Theorem 2.2. Let G be a distance-regular graph whose intersection array is {b0 , b1 , ..., bD−1 ; c1 = D−1 nD−1 bD 0

1, c2 , ..., cD }. Then we have Wλ∗ (G) =

∏ bD−i i i=1

D!λ .

D 4



cD+1−i i

i=2

Theorem 2.3. Let G be a distance-regular graph of diameter D with n vertices. Then for each u ∈ V (G) D

D(u) = ∑ iki , i=1 D

W (G) =

n ∑ iki , 2 i=1

D

D∗ (u) = D! ∏ ki , i=1

W ∗ (G) =

D!n D n ki . 2 ∏ i=1

A general case of the above theorem is formulated in the following statement whose proof is done in the same way of Theorem 2.3. Theorem 2.4. Let G be a distance-regular graph of diameter D with n vertices. Then for each n D λ λ u ∈ V (G) Dλ (u) = ∑D i=1 i ki , Wλ (G) = 2 ∑i=1 i ki . Theorem 2.5. Let G be a distance-regular graph of diameter D with n vertices. Then H(G, x) = n D i 2 ∑i=1 ki x .

153

Theorem 2.6. Let G be a bipartite distance-regular graph of diameter D with n vertices. Then for each f = uv ∈ E(G) D−1

D( f ) =



 iki bi + (i − 1)b1 b2 . . . bi ,

We (G) =

i=1

 nb0 D−1 ∑ iki bi + (i − 1)b1 b2 . . . bi . 4 i=1

The line graph L(G) of a graph G is defined as follows: each vertex of L(G) represents an edge of G, and any two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint in G. One can also define iterated line graphs by setting L0 (G) = G, L1 (G) = L(G) and generally Ln (G) = L(Ln−1 (G)). The following observation is due to M.H. Khalifeh at. el [6, Theorem 2.4] Theorem 2.7. Suppose G is a connected graph with m edges. Then   m W (L(G)) −We (G) = . 2 Corollary 2.8. Let G be a bipartite distance-regular graph of diameter D with n vertices. Then W (L(G)) = We (G) +

     nb0 /2 nb0 /2 nb0 D−1 ik b + (i − 1)b b . . . b + = i 1 2 ∑ ii 4 i=1 2 2

References [1] N. Biggs, Algebraic Graph Theory, 2nd ed., Cambridge University Press, Cambridge, 1993. [2] A.E. Brouwer, A. M. Cohen, A. Neumaier, Distance–Regular Graphs, Springer–Verlag, Berlin, 1989. [3] A.A. Dobrynin, I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math. Beograd 56 (1994) 18–22. ˇ Tomovi, The Multiplicative Version of the Wiener [4] I. Gutman, W. Linert, I. Lukovits and Z. Index, J. Chem. Inf. Comput. Sci., 40 (1) (2000), 113–116. [5] H. Hosoya, On some counting polynomials in chemistry, Discr. Appl. Math. 19 (1988), 239– 257. [6] M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distancebased graph invariants, European J. Combin., 30 (2009)1149–1163. [7] J. A. Rodr´ıguez, On the Wiener index and the eccentric distance sum of hypergraphs, MATCH Communications in Mathematical and in Computer Chemistry 54 (1) (2005) 209–220. [8] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69(1974) 17–20.

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