Study of Circular Distance in Graphs

Abstract: Circular distance between vertices of a graph has a significant role, which is defined as summation of detour distance and geodesic distance. Attention is paid, this is metric on the set of all vertices of graph and it plays an important role in graph theory. Some bounds have been carried out for circular distance in terms of pendent vertices of graph . Some results and properties have been found for circular distance for some classes of graphs and applied this distance to Cartesian product of graphs〖 P〗_2×C_n. Including 〖 P〗_2×C_n, some graphs acted as a circular self-centered. Using this circular distance there exists some relations between various radii and diameters in path graphs. The possible applications were briefly discussed.


Introduction
Due to the broad expansion of networks in graph theory, the distance concept has become very important. It has been recognized that the distance concept in graph theory is also useful in software development. Using this distance concept some basic graph theory parameters were defined such as eccentricity, radius, diameter and metric dimension. The above parameters are related to one another and some bounds have been derived for the radius and diameter [1]. The geodesic distance ( , ) d  [2 -4]. Sometimes calculating degree of each vertex of a graph H is also important. In that case in addition to the length of the path, consider degree of each vertex in that path. So many real life problems were solved using this graph distance. For example, when a van is traveling to deliver the goods, it has to stop at one point and deliver the goods: consider the point as vertex and the number of goods delivered as degree of that vertex. Similarly it has to travel and deliver the goods at all delivery points, by taking it as a detour D − distance and return to home place, taking this distance as D − distance. By doing this type of work, it helps to save time and fuel usage [5,6]. Now a days wireless network is rapidly growing, in this type of network for example radio, mobile etc., are very useful. Assignment of frequencies to transmitters depend on distance between two transmitters [7 -9]. On the off chance that L and M are two urban areas, at that point for a cab driver the distance between the two urban communities is the genuine distance between the two urban areas. Anyway for a bus transport driver, the distance between similar urban areas is only higher than the typical distance since he needs to cover some significant places in and around the two urban communities to get and drop the travelers. So a bus transport driver needs to locate a briefest defeat that starts from L and closures at M and goes through every one of the neighboring spots of L and M [10][11][12][13]. In this paper we study a new distance, namely circular distance, between any two vertices of a graph H by considering summation of detour distance and geodesic distance and work on its properties The circular distance takes part in a vital role in logistical management. For instance, a postman wants to distribute the letters from the post office which is, say, located at a place 'A' to various destinations on the way to another place 'B'. He takes a long trip from A to B so that he can cover as many as places on the way. On the return trip he may select the shortest route to arrive at the post office at the earliest so as to minimize the time, fuel consumption etc. Similarly, in case of delivery of goods from warehouse to various places we may choose longest path and shortest path. This distance was studied in [14]. Further more, this distance is also used in molecular dynamics of physics, crystallography, lattice statistics and physics. Throughout the article unless otherwise specified we consider undirected, finite, non-trivial, connected graphs without multiple edges and loops. For any unexplained terminology we follow the book [1].

Circular distance and its properties
Here we explain the concept of circular distance between vertices of a graph: Proof: Let H be a graph with pq + vertices and assume that among them q are pendent vertices. For any two vertices , xy which are not pendent and 12 , , , q x x x be all the pendent vertices. Any xy − path will exclude these vertices and the edges incident with them. Hence ( ) ( ) Suppose one vertex between x and y (say x ) is a pendent vertex. Then each xy − path will begin at .
x These paths will go through the edge adjacent with .
x They will not go through the remaining 1 q − vertices and edges incident with them. Thus each of these minimum distances will have a maximum of 1 pq −+ edges.
For pendent vertices x and , y the minimal xy − path starts at x and ends at . y This minimal path passes through the edges adjacent with both x and y does not go through the rest of 2 q − pendent vertices.
Proof: The lower bound is true. By using triangular inequality the upper bound is as shown below. Let ( ) , x y V H  such that circular distance between x and y is equal to circular diameter of .

Circular distances of some classes of graph
Here we find the circular diameter and circular radius. We start with the family of complete graphs. Proof. In m K , the minimum path length between any two vertices is 1 and maximum length of path is Next, we go for class of cycle graph x is adjacent to 1 2 3 , , .
x x x Then clearly ( ) Hence the circular eccentricity of every vertex is 4. Thus For a wheel graph 1, , n W with 4 n  , we calculate the circular distance between 0

Trees with respect to circular distance.
Here we study the some results on trees.
Proof. The vertices of path graph n P can be listed in order 12 , , , n x x x such that the edges are   1 ii xx + where 1, 2, 3, , 1. in =− There is one and only path between two vertices and hence detour distance and geodesic distance are equal. Now for 2 n = , Now let us calculate the radius and diameter of n P for n3  using circular distance. This we will do when n is even and odd. Case (1): Assume n is even.
Here the circular distances between the vertices of n P are as shown in table 5.
Table5. circular distance of path graph when n is even    Proof. We can demonstrate this hypothesis in two cases in particular when n is even and odd.
If n is even then center of n P consist of two adjacent vertices. From above theorem Next in a star graph, circular diameter and radius are independent of n .   Proof: Let T be a tree. Then between any two vertices the maximum circular distance occurs then they must be end vertices. Thus if a vertex has maximum circular distance then that vertex must be in the circular periphery. Hence the circular periphery of a tree contains pendent vertices only. Some of the future applications.
Below we discuss some possible applications of the concept of circular distance in real life. Here we discuss three of such applications. Delivery of goods using drones . Now a days some of the delivery companies are using the drones to deliver the goods. The drones will have the option to fly up to 15 miles and convey bundles under five pounds to clients in under $30$ minutes time. For instance the Amazon prime company used this for the first time. At present it is using the one to one correspondence procedure to deliver goods. In this method it will take more cost and time. To reduce the cost and time we can use this new concept of circular distance. Using this concept, when we feed the delivery points to the system, the drone will choose the detour path containing more delivery points and choose the shortest path to come back.

Automatic map generation of driverless car with safe journey
In future we will have most updated driverless cars. While going on a tour using these cars, we have to set the destination and visiting places. Then automatically car will go through these places by using some algorithm. Suppose when we use this circular distance concept in these cars, the tour will go smoothly with low cost and can be reduced time also. Considering the vertices as visiting places and giving this information to the car system by using circular distance concept it will generate the map which consists of detour path containing more visiting places and shortest path while coming back home. Thus we will reduce the time and cost of the tour.

V2V communication in 5G technology
In driverless vehicles, there must be communication between vehicles. For instance while traveling, generally we come across signals in junctions. Near the signals the car has to stop and communicate to the vehicles behind it, that ahead there is a red signal and reduce the speed of the car. This message has to go to the vehicles behind up to certain number of cars depending upon the programming. Again the destination car has to send signals to the source car that it received the message. Thus all the cars reduce their speed automatically. When the source car communicates the message it will choose the detour path and destination car sends the message using the shortest path. Indirectly here the mechanism is circular distance. Here the cars are vertices. Using this mechanism we can reduce accidents of the cars in traffic and we will have more peaceful and safe journey.

Conclusion
In this work, we studied circular distance in a graph and we proved that circular distance is metric. We concentrate few properties of circular distance. We found the circular distance of some families of graphs. We proved that few of the graphs are self-centered. Further we proved 2 × is also a self-centered graph. In future work, we want to study radio circular distance between edges of a graph and also find the relation between circular distance and radio circular distance of a graph. Further we will study the properties of radio circular distance of a graph.