International Journal of Advanced Engineering Technology E-ISSN 0976-3945 Research Article DISTANCE IN GRAPH THEORY AND ITS APPLICATION Mahesh C. Prajapati Address for Correspondence Assistant Professor, Shree Saraswati Education Sansthan’s Group of Institutions, At & Po.: Rajpur, Ta. Kadi, Dist. Mehsana ABSTRACT I will be define the distance partition of a finite graph and show how this partition being equitable follows from and/or implies properties of the graph. In the process I will connect this partition to a number of fundamental ideas in graph theory and confirm an elementary identity of strongly regular graphs. KEYWORDS Length of a path, Distance in graph theory, Eccentricity, Radius and Diameter of a graph, Center vertex, Center of a graph. INTRODUCTION We may write G = (V (G), E (G)). We see that how a graph can be used to model the street Definition:- Vertices & Edges system of a town. Of course, as a town grows in size, so If G = (V(G) , E(G)) is a graph then the set V(G) is said too does not the graph at model it. As a reminder, we to be the vertex set of the Graph G and the member of see in below figure the street system of a town T and a V(G) are said to be the Vertices of the Graph. The Graph G T that model it. family E (G) is said to be the age family of the graph (G) and the members of E (G) are said to be the edges of the Graph G. Example: Let V={1,2,3,4,5} & E = { (1,1),(1,3),(1,2),(1,2), (1,4),(2,5),(2,3),(3,4),(4,5),(5,5) } then G = (V,E) is a Graph. Here V(G) = V is the vertex set of G and 1,2,3,4,5 are the vertices and E(G) = E is edge family of G and (1,1), Town T GT: Graph modeling (1,3), (1,2) twise, (1,4), (2,5), (2,3), (3,4), (4,5), (5,5) Town T are edges. For example when a town is small, it might be This graph may be represented by the following appropriate to rely on and pay for the services of the diagram. fire department of a neighboring city. As a town grows in to a city (and it able to affords It.), new question arise, it becomes necessary for that town to have its own fire department. Assuming that the decision has been made by the town to build its own firehouse, we now have another question: Where in the town should we build it? Let’s assume that we decide to build the firehouse at Definition:- A Simple Graph some street intersection in the town. A simple Graph (G) consists of a nonempty finite set V This how much, doses not answer our question. Of (G) together with a set E(G) of unordered pairs of course, the main reason for building the firehouse is so distinct element of the set V(G). that all citizens of the town are protected in the event of We may write G = (V (G), E (G)). the fire. Example: Consequently, no location in the town should be too far Let G = (V(G),E(G)) where V = V(G) = {a, b, c, d, e, f} from this new firehouse. and E = E(G) = { ab, bc, cd, ef } then G is a simple We see that answering our question concerns distance Graph. in town T and there for Distance in the graph GT as well. For this first we define distance & some basic definitions of a Graph Theory. Definition:- A Graph or a general Graph A Graph (G) or a general Graph (G) consists of a nonempty finite set V (G) together with a family E(G) of unordered pairs of element (not necessarily distinct) of the set. IJAET/Vol.II/ Issue IV/October-December, 2011/147-150 International Journal of Advanced Engineering Technology E-ISSN 0976-3945 Definition:- A walk The eccentricity e (v) of a vertex v of a connected graph A walk in a graph G is a finite sequence of edges of the G is the number form v v , v v , v v . V v . Also denoted by v → 0 1 1 2 2 3 m-1 m 0 Definition: - Radius of a graph G v → v → . → v → v , in which any two 1 2 m-1 m The radius of a graph G is denoted by rad G and is consecutive edges are adjacent or identical. Such a walk defined as rad G = determines a sequence of vertices v 0, v 1, v 2, . , v m-1, and v m. (Here two consecutive vertices are adjacent or Definition: - Diameter of a graph G identical). The diameter of a graph G is denoted by Here v 0 is said to be initial vertex of the walk & v m is diam G = said to be the terminal vertex of the walk. It follows that diam G = Definition: - Length The no. of edges in a walk is said to be the length of the Definition: - Center of graph G walk. i.e. the length of the walk is the no. of term of A vertex v is a central vertex if e (v) = rad G. that sequence . And the center of G consists of its central vertices. The Definition: - Trail radius & diameter are related by the following If all the edges in a walk are distinct then such a graph inequality. is said to be a Trail. Theorem: - For any connected graph G, rad (G) ≤ Definition: - Path diam (G) ≤ 2 rad (G). Proof:- The inequality rad(G) ≤ diam (G) is direct If all the vertices v 0, v 1, v 2. v m in a trail are distinct consequences of the definitions since the smallest (except possibly v 0 = v m) then such a trail is said to be a path . eccentricity can not exceed the largest eccentricity. Definition: - A walk or a Trail or a Path is said to be a There fore rad (G) ≤ diam (G) …… (1) close if initial vertex & the final vertex are identical. In order to verify the second inequality select vertices u Definition: - The Union of Graphs. & v in graph G such that d (u, v) = diam G. Further more, let w be a central vertex of G If G 1 = (V(G 1), E(G 1)) & G 2 = (V(G 2), E(G 2)), where i.e. e (w) = rad (G) V(G 1) ∩ V(G 2) = ф . Then their union is denoted by Since d is a metric on V (G). G1U G 2 & it is the graph with the vertex set V(G 1) U d (u, v) ≤ d (u, w) + d (w, v) ≤ rad (G) + rad (G)= 2 rad V(G 2) & the edge family E(G 1) U E(G 2). Definition: - A connected Graph (G) A Graph is said to be a connected graph if it can not to =>d (u, v) ≤ 2 rad (G) be expressed as the union of two Graphs. i.e diam (G) ≤ 2 rad (G). …….. (2) Definition: - Distance Hence (1) & (2) => rad (G) ≤ diam (G) ≤ 2 rad (G). For two vertices u &v in a graph G, the distance from u Definition: - Adjacent vertices to v is denoted by d (u, v) & defined as the length of a Two vertices u & v of a Graph G are said to be shortest u-v path in graph G. Adjacent vertices if there exists an edge joining them. For connected Graph G the term distance we just And we say that the vertices u &v are incident with the defined satisfies all four of the following properties. edge uv. 1) d (u, v) ≥ 0 for all u, v € V(G). Theorem:- 2) d (u, v) = 0 iff u = v. For every two adjacent vertices u & v in a connected 3) d (u, v) = d (v,u) for all u, v € V(G) (The graph, | e(u) – e(v) | ≤ 1. symmetric property). Proof:- Assume, without loss of generality, that e (u) ≥ 4) d (u, w) ≤ d(u, v) + d(v, w). for all u, v € e(v). V(G).(The triangle inequality). Let x be a vertex that is farthest from u. So, d (u, x) = e (u). If graph G is not a connected & suppose G 1 & G 2 are two graphs of G such that By the triangle inequality, e(u) = d(u, x) ≤ d(u,v) + d(v, x) G = G 1 U G 2 & E (G 1) U E (G 2). ≤ 1 + e (v) & G1∩ G2 = ф i.e. V (G 1) ∩ V (G 2) = ф & Hence e (u) ≤ 1 + e (v) E (G 1) ∩ E (G 2) = ф =>0 ≤ e (u) – e (v) ≤ 1 Then d (u, v) = ∞ for u € V (G 1) & v € V (G 2). Under this distance function, the set V(G) is a metric There fore | e (u) – e (v) | ≤ 1. space & it is denoted by (V(G),d ) , there are several Theorem: - Every Graph is the center of some graph. references which we shall make to this distance Proof:- Let G be a Graph. We shall show that G is the function, however in this section we describe some center of some graph. concepts which are intimately related to distance and First, add two new vertices u & v to G & join them to which are of interest in their own right. every vertex of G but not to each other.