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. Definition:-Eccentricity of a Vertex V in graph G Next, we add two other vertices u 1 &v 1, where we join u1 to u & join v 1 to v.

IJAET/Vol.II/ Issue IV/October-December, 2011/147-150

International Journal of Advanced Engineering Technology E-ISSN 0976-3945

The resulting Graph is denoted by F. Definition: - Eccentric vertex If e (u) = d (u, v); where u, v € G, i.e. v is a vertex that is farthest from u. Such a vertex v is called an eccentric vertex of u. A vertex v is an eccentric vertex of the graph G if v is an eccentric vertex of some vertex of G. Since e (u 1) = e (v 1) = 4, e (u) = e (v) = 3, & e(x) = 2 In other words, a vertex v is an eccentric vertex of G v for every vertex x in G. is farthest from some vertex of G. It shows that V (G) is the set of central vertices of F & Definition: - Eccentric Graph so, Cen (F) =G A connected graph G has the same eccentricity (& is Hence the proof that every graph is a center of sub there fore a peripheral vertex), then certainly every graphs. vertex of G is an eccentric vertex. Definition: - Peripheral vertex A connected graph G is an eccentric graph if every A vertex v in a connected Graph G is called a vertex of G is an eccentric vertex. peripheral vertex if e (v) = diam (G). Every vertex of a graph can be an eccentric vertex Thus in a certain cense, a peripheral vertex is opposite without all the eccentricities being the same. to a central vertex. The sub graph of G induced by peripheral vertices is the periphery. This is denoted by per (G). Example:-

rad (H) = 2 & diam (H) = 4 Example:- A graph whose vertices in an eccentric vertex which are labeled by its eccentricity . Definition: - Eccentric sub graph’ Let G be a connected graph. The eccentric sub graph Ecc (G) of G is a sub graph of G induced by the set of eccentric vertices of G. Definition: - Separating Set

Question:- Is the periphery of every graph A separating set in a connected graph G is a set of disconnected? vertices whose deletion disconnects G. Answer: - NO. Definition: - Cut-vertex For this we take an example. If a separating set of G contains only one vertex v, then the vertex v is said to be a cut-vertex of G. Definition:- Boundary vertex A Vertex is in a connected graph G is a boundary vertex of a vertex u if d (u, w) ≤ d (u, v) for each neighbor w of v. While a vertex v is a boundary vertex of the graph G if u is a boundary vertex of some vertex of G. Note: - If v is an eccentric vertex of a vertex u in a connected graph G, then no vertex of G is farther from u than v is. In particular, if w is a neighbor of v, then d (u, w) ≤ (u,v). The graph F of above figure shows, each vertex of F is Theorem: - No cut vertex of a connected graph G is labeled with its eccentricity. a boundary vertex of G. Since diam (F) = 3, it follows that per (F) ≈ C , which is 6 Proof: - Assume, to the contrary, that there exists a connected. connected graph G and a cut-vertex v of G such that v

is a boundary vertex of some vertex u in G, Let G 1 be component of G-v that contains u and let G 2 be another component of G-v. If w is a neighbor of v that belongs to G2, then d(w ,u) = d(u,v) +1,

IJAET/Vol.II/ Issue IV/October-December, 2011/147-150

International Journal of Advanced Engineering Technology E-ISSN 0976-3945

Which contradicts our assumption that v is a boundary see that an appropriate answer is for the fire house to be vertex of u. Because if v is a boundary vertex if u then built at any of the four intersection correspond to d(u, w) ≤ d(u, v), if w is a neighbor of v & here d(w, u) central vertices (vertices having eccentricity 4) in GT. ≥ d(u, v). Therefore if v is a cut-vertex of a connected graph G then it is a boundary vertex of G is false. Hence the proof. Note: - Since no cut-vertex can be a boundary vertex, no cut-vertex can be an eccentric vertex or a peripheral vertex either. There are certain vertices, however, that must be boundary vertices. Definition: - Interior vertex of graph G A vertex v is an interior vertex of G if for every vertex u distinct from v, there exists a vertex w such that v lies between u and w. DISCUSSION The interior Int (G) of G is the sub graph of G induced Here we discussed on very simple model of town But if by interior vertices. there is any complicated model for example, suppose if there is any river passing through that town then we can also discuss here the same problem with the help of concept of bridge in graph theory and it can also be solved easily. Graph theory is very powerful, Simple & also very interesting topic to work with any physical and real life problem. ACKNOWLEDGMENTS I feel immensely proud to present this report consisting For Example, of mathematical logics and computing skill. This work For the graph G of above figure the vertices s, v & x are thought me not only to be patient but also made me the interior vertices of G and so Int (G) = P , as 3 eager to venture into new heights of taking up logics showing in above figure. into been possible without outside help. Hence, I would We now see that the interior vertices are precisely those like to thanks my parents and Dr. Snehal B. Rao vertices that are not boundary vertices. (Assistant professor of Applied Mathematics Theorem: - Let G be a connected graph, A vertex v department in M.S. University, Baroda) for good is a boundary vertex of G iff v is not an interior support. Thank you all. vertex of G. REFERENCE Proof:- Let v be a boundary vertex of G, say v is a 1. Graph Theory and its Applications by Gross.JL and boundary vertex of the vertex u. Assume, to the Yellen.J CRC Press LLC, 1998 contrary, that v is an interior vertex of G. Since v is an 2. Graph Theory by Diestel.R Springer-Verlag 1997 interior vertex of G, there exists a vertex w distinct 3. Introduction to Graph Theory by West.DB Prentice from u & v such that v lies between u & w. Hall 1996 4. Introduction to Graph Theory by R.J. Wilson Addison Let P: u = v 1, v 2, v 3,….. , v = v j, v j+1 , … , v k = w be a u- Wesley Longman 1996 v path, Where 1< j < k. However, v j+1 € N (v) and 5. Introduction to Graph Theory by R.J. Trudeau Dover d(u,v j+1 ) = d(u, v) + 1, a contradiction. Pubns 1994 For the converse, let v be a vertex that is not an interior 6. Graphs: An Introductory Approach by J. Wilson and vertex of G. Hence there exists some vertex u such that J.J. Watkins John Wiley & Sons 1990 7. Graph Theory by Gould.RJ Benjamin/Cummings 1988 for every vertex w distinct from u & v, the vertex v 8. Introduction to Graph theory By Gary chartrand & does not lie between u & w. Let x € N(v),then Ping Zhang. d (u, x) ≤ d (u, v) + d(v, x) = d(u, v) +1. Since v does not lie between u &x, this inequality is strict and so d (u, x) ≤ d(u, v), that is, v is a boundary vertex of u. Hence the proof. CONCLUSION

The graph G T of figure 1, (Which recall, model town T is that figure) is so again in figure 3, Where in this case every vertex is labeled with its eccentricity. We ask earlier where a fire house be built so that no location in the town is too far from fire house. We now

IJAET/Vol.II/ Issue IV/October-December, 2011/147-150