DIAMETRAL PATHS IN TOTAL GRAPHS OF COMPLETE GRAPHS, COMPLETE BIPARTITE GRAPHS AND WHEELS

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1 International Journal of Civil Engineering and Technology (IJCIET) Volume 8, Issue 5, May 017, pp , Article ID: IJCIET_08_05_17 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed DIAMETRAL PATHS IN TOTAL GRAPHS OF COMPLETE GRAPHS, COMPLETE BIPARTITE GRAPHS AND WHEELS Tabitha Agnes Mangam Department of Mathematics, Christ University, Bengaluru, Karnataka, India Joseph Varghese Kureethara Department of Mathematics, Christ University, Bengaluru, Karnataka, India ABSTRACT The diametral path of a graph is the shortest path between two vertices which has length equal to diameter of that graph. In raising of structures with columns and beams in Civil Engineering, determining of diametral paths is of great significance. In this paper, the number of diametral paths is determined in complete graphs, complete bipartite graphs, wheels and their total graphs. Key words: Diameter, Peripheral vertex, Central vertex, Diametral path, Total graph Cite this Article: Tabitha Agnes Mangam and Joseph Varghese Kureethara. Diametral Paths in Total Graphs of Complete Graphs, Complete Bipartite Graphs and Wheels. International Journal of Civil Engineering and Technology, 8(5), 017, pp INTRODUCTION Columns and beams make a structure. A structure can be represented by a graph. Edges are parts or the full of the steel bars inside columns and beams in large and small scale structures. A. Kaveh skillfully combines concepts of graph theory and matrix algebra to present powerful tools for the analysis of large-scale structures as seen in [6]. To minimize the wastage of steel bars inside the concrete columns and beams in large-scale structures, structural engineers usually go for maximizing the length of the steel bars. This also increases the load-bearing capacity of the structure in reference to [9] and [10]. Diametral path in a graph is the longest induced path in a graph. Identifying the diametral paths in a structure will help us in optimally using the steel bars. In this paper, a study of diametral paths in certain graph structures like total graphs is undertaken. Diameter and diametral paths are discussed in detail in literature [[1], [3], [8]] and [4]. In Section, the number of diametral paths in total graph of complete graphs is determined. In 11 editor@iaeme.com

2 Diametral Paths in Total Graphs of Complete Graphs, Complete Bipartite Graphs and Wheels Section 3, diametral paths in total graphs of complete bipartite graphs are investigated. In section 4, total graphs of wheels are discussed in the same context. The definitions and results are in reference to [[],[5]]. The degree d(v) of a vertex v of a graph G is the number of edges incident on v. The length of a path is the number of edges on the path. The distance d G (u, v) or d(u, v) between two vertices u, v V(G) in a graph G is the length of shortest path between them. The eccentricity of a vertex is the maximum of distances from it to all the other vertices of that graph. While diameter diam(g) of a graph G is the maximum of the eccentricities of all vertices of that graph, the radius is minimum of these. Peripheral vertices are vertices of maximum eccentricity and central vertices are of minimum eccentricity. Complete graph K n is a graph with n vertices and an edge between every pair of vertices. Bipartite graph is a graph in which the set of vertices are partitioned into two subsets V 1 and V such that every edge has one end vertex in V 1 and the other in V. Complete bipartite graph K r,s is a bipartite graph in which there is an edge from every vertex of V 1 to every vertex of V. Cycle C n is a graph with n vertices {v 1, v,,v n } and n edges namely {(v 1, v ); (v, v 3 ),,(v n-1, v n )} and an additional edge (v n, v 1 ). Wheel W n is a graph of n vertices formed by connecting a single vertex to the n-1 vertices of a cycle C n-1. The diametral path of a graph is the shortest path between two vertices which has length equal to diameter of that graph. The vertices and edges of a graph are called its elements. Two elements of a graph are neighbors if they are either incident or adjacent. The total graph T(G) as introduced by F. Harary [5] has vertex set V (G) E(G) and two vertices of T(G) are adjacent whenever they are neighbors in G. There is an edge between two vertices in T(G) if and only if there is edgeedge adjacency or edge-vertex incidence or vertex-vertex adjacency between the corresponding elements in G. Hence the number of vertices in T(G) is n+m where n are vertices of the original graph G(n, m) and m are the new vertices representing edges of the original graph. The structure of total graphs is discussed by J. Thomas and J. Varghese in [7].. TOTAL GRAPHS OF COMPLETE GRAPHS In this section, the diametral paths are analysed in a complete graph and its total graph..1. Proposition A complete graph K n (n ) has diametral paths. Proof. Since diam(k n ) = 1, every edge is a diametral path. Since there are edges, there are diametral paths. Hence the number of diametral paths in a complete graph K n is... Theorem The diameter of the total graph of a complete graph K n, diam(t(kn) = and the number of diametral paths in T(Kn)is ()( ). Proof. Consider the vertices of K n. Since diam(k n ) = 1 and vertices are mutually adjacent in K n, they remain mutually adjacent as vertices in T(K n ). Hence distance between any two of them remains to be 1 in T(K n ). Consider a vertex u 1 and its non incident edge e 1 in K n as seen in its subgraph in Figure 1. Since the end vertices of e are adjacent to u, these are edges e and e 3 from u 1 to end vertices u and u 3 of e 1 respectively in K n. In T(K n ), the shortest paths between u 1 and e 1 are of length and are u 1 u e 1, u 1 e e 1, u 1 u 3 e 1 and u 1 e 3 e 1. Hence the diam(t(k n ) =. There are 4 diametral paths from each vertex of K n to its non incident edge as vertices of T(K n ). Since there are editor@iaeme.com

3 Tabitha Agnes Mangam and Joseph Varghese Kureethara -(n-1) non incident edges to each vertex of n vertices of K n, the number of diametral paths in T(K n ) of this kind is n [ -(n-1)] 4. Figure 1 Subgraph H 1 of K n Consider any two non adjacent edges e 4 and e 5 of K n as seen in its subgraph in Figure. Let the end vertices of e 4 be u 4 and u 5. Let the end vertices of e 5 be u 6 and u 7. In K n, u 4, u 5, u 6 and u 7 are adjacent to each other. Hence there are 4 edges e 6, e 7, e 8 and e 9 between them as seen in Figure. In T(K n ), there are 4 shortest paths between e 4 and e 5 of length which are e 4 e 6 e 5, e 4 e 7 e 5, e 4 e 8 e 5 and e 4 e 9 e 5. Hence there are 4 diameteral paths between any two non adjacent edges of K n as vertices of T(K n ). Since there are -(n-1+n-) non adjacent edges to each edge of K n, the total number of diametral paths in T(K n ) of this kind without repetition is [ -(n-1+n-)] 4. Hence total number of diametral paths is (n [ - (n-1)] 4)+( [ -(n-1+n-)] 4 ). diametral paths in T(K n ) as ()( ). Simplifying this, we get the total number of Figure Subgraph H of K n 3. TOTAL GRAPH OF COMPLETE BIPARTITE GRAPHS 3.1. Proposition The graph K m, n (m or n ) has m +n diametral paths. Proof. Let V 1 = m and V = n. Since diam(k m, n ) = and every vertex in V 1 has diametral paths passing through it, there are m diametral paths through the vertices of V 1. Similarly, there are n diametral paths through the vertices of V editor@iaeme.com

4 Diametral Paths in Total Graphs of Complete Graphs, Complete Bipartite Graphs and Wheels Hence the total number of diametral paths in a complete bipartite graph K m, n (m or n ) is m +n. 3.. Theorem The diameter of the total graph of a complete bipartite graph, diam(t(k m, n )) = and the number of diametral paths in T(K m, n ) is mn(3m + 3n + mn - 8)/. Proof. Let V 1, V be the partite sets of K m, n. Let V 1 = m and V = n. Since diam(k m, n ) = and the vertices of each partite set are non adjacent among themselves in K m, n, there are such pairs of non adjacent vertices in V 1. Also there are such pairs in V. Since distance between any two non adjacent vertices in K m, n is, it remains the same in T(K m, n ). In T(K m, n ), the shortest paths between any two vertices of a partite set of K m, n can pass through any of the vertices of the other partite set. Hence the number of such shortest paths are of length and are ( n)+( m) in number. Figure 3 Subgraph H 1 of K m, n Next, we consider any two non adjacent edges e 1 and e of K m, n as seen in its subgraph in Figure 3. Let end vertices of e 1 be u 1 and u in K m, n. Let end vertices of e be u 3 and u 4 in K m, n. Let u 1 and u 3 belong to V 1. Let u and u 4 belong to V. In K m, n there are edges e 3 from u 1 to u 4 and e 4 from u to u 3. Hence the shortest paths from e 1 to e in T(K m, n ) are of length and are e 1 e 3 e and e 1 e 4 e. For each edge, the number of its non adjacent edges is mn-(m + n -1). Hence the total number of shortest paths of this kind without repetition is (mn(mn-(m+n- 1)) )/. Figure 4 Subgraph H of K m, n Consider a vertex u 5 of V 1 and any non incident edge e 5 in K m, n as seen in its subgraph in Figure 4. Let the end vertices of e 5 be u 6 and u 7 where u 6 belongs to V 1 and u 7 belongs to V. In K m, n, there is an edge e 6 from u 5 to u 7. Hence in T(K m, n ), the shortest paths from u 5 to e editor@iaeme.com

5 Tabitha Agnes Mangam and Joseph Varghese Kureethara are of length and are u 5 u 7 e 5 and u 5 e 6 e 5. For each vertex of V 1, there are (m-1) n non incident edges. Also for each vertex of V, there are (n-1) m non incident edges. Hence the shortest paths of this kind are of length and ((m -1) n) + ((n-1) m) in number. Hence diam(t(k m, n )) = and the total number of diametral paths in T(K m, n ) is + (()) ) + + ( 1) + ( 1). Simplifying, we get mn(3m + 3n + mn - 8)/. 4. TOTAL GRAPH OF WHEELS 4.1. Proposition A Wheel W n (n 5) has diametral paths. Proof. Since diam(w n ) = when n 5, there are n - 1 peripheral vertices in W n. Let v1, v, v 3 v n-1 be the n-1 peripheral vertices of W n. Considering the diametral paths from every vertex v i through v i+1, 1 i n- and one from v n-1 through v 1, there are n-1 diametral paths through them. Also between the n-1 peripheral vertices there are paths of length through the central vertex. Eliminating the n - 1 paths between the adjacent peripheral vertices, we get ( 1) diametral paths through the central vertex. Hence the total number of diametral paths is ( 1) + ( 1). Simplifying, we get. 4.. Theorem For the wheel graphs, (1) diam(t(w 4 )) = and the number of diametral paths in T(W 4 ) is 60. () diam(t(w n )) = 3 for n 7 and the number of diametral paths in T(W n ) is (n - 1)(n - 35). Proof. (1) Consider W 4. SinceW 4 is K 4, as discussed earlier we get diam(t(w 4 )) = diam(t(k 4 )). Hence the number of diametral paths in T(W 4 ) is the number of diametral paths in T(K 4 ). That is [4(4-1)(4-4 - )]/= 60. () Consider W n for n 7. Let C be the central vertex of W n. Since C is adjacent to the remaining vertices of W n, they are all at a distance of not more than from each other in T(W n ). Figure 5 Subgraph H 1 of W n editor@iaeme.com

6 Diametral Paths in Total Graphs of Complete Graphs, Complete Bipartite Graphs and Wheels Consider a peripheral vertex u 1 and an edge e 1 which is not incidenton u 1 or any of its adjacent vertices as seen in a subgraph of W n in Figure 5. Let end vertices of e 1 be u and u 3. Since u 1, u and u 3 are all adjacent to C, there are edges e, e 3 and e 4 respectively from those vertices to C. Hence the shortest paths from u 1 to e 1 in T(W n ) are of length 3 and are u 1 Cu e 1, u 1 Cu 3 e 1, u 1 Ce 3 e 1, u 1 Ce 4 e 1, u 1 e e 3 e 1 and u 1 e e 4 e 1. Hence, diam(t(wn)) = 3 and there are 6 diameteral paths of this kind. Suppose u 1 and u 3 are adjacent to a common peripheral vertex u 4 with respective edges e 5 and e 6 as seen in a subgraph of W n in Figure 6. Then there are 3 additional shortest paths from u 1 to e 1 in T(W n ) of length 3 which are u 1 u 4 u 3 e 1, u 1 e 6 e 5 e 1 and u 1 u 4 e 5 e 1. If u 1 and u are adjacent to another peripheral vertex in W n, there would be 3 additional shortest paths between u 1 and e 1 in T(W n ). Since each vertex of W n has (n-1)-4. i.e., n-5 of the specified kind of edges, the number of diametral paths of this kind in T(W n ) is (n-1) (3+((n-5) 6)+3). i.e., 6(n-1)(n-4). Figure 6 Subgraph H of W n Next, we consider any two non adjacent edges e 1 and e which have non adjacent peripheral vertices u 1, u, u 3 and u 4 respectively as end vertices in W n as seen in a subgraph of W n in Figure 7. Since u 1, u, u 3 and u 4 are all adjacent to C, there are edges e 3, e 4, e 5 and e 6 respectively from them to C in W n. Hence the shortest paths from e 1 to e in T(W n ) are of length 3 and are e 1 e 3 e 5 e, e 1 e 3 e 6 e, e 1 e 4 e 5 e and e 1 e 4 e 6 e. Hence, there are 4 diametral paths of this kind. Figure 7 Subgraph H 3 of W n Suppose u 1, u 4 the end vertices of e 1 and e respectively are adjacent to a common peripheral vertex u 5 in W n as seen in a subgraph of W n in Figure 8. Then there is an additional shortest path in T(W n ) from e 1 to e of length 3 which is e 1 e 8 e 9 e. If the other pair of end editor@iaeme.com

7 Tabitha Agnes Mangam and Joseph Varghese Kureethara vertices of e 1 and e are adjacent to a common peripheral vertex in W n, then there is an additional shortest path of length 3 from e 1 to e in T(W n ). Figure 8 Subgraph H 4 of W n Since each edge of W n with peripheral end vertices has (n - 1) 5. i.e., n - 6 edges of the specified kind, the number of diametral paths of this kind in T(W n ) without repetition is ((n- 1)/) (1+((n-6) 4)+1). i.e., (n - 1)(n - 11). Hence the total number of diametral paths in T(W n ) is 6(n - 1)(n - 4) + (n - 1)(n-11). Simplifying, we get (n - 1)(8n - 35) Corollary For a wheel graph, diam(t(w 6 )) = 3 and the number of diametral paths in T(W 6 ) is 60. Proof. Since W 6 has no pair of non adjacent edges with non adjacent peripheral vertices as end vertices, the number of diametral paths in T(W 6 ) is 6(n - 1)(n - 4). As n is 6, we get it as CONCLUSION In this paper, the number of diametral paths is determined in complete graphs, complete bipartite graphs, wheels and their total graphs. The focus of further research would be the investigation of diametral paths related parameters in total graphs of these classes of graphs. REFERENCES [1] Bermond, J. C., Bond, J., Paoli, M. and Peyrat, C. Graphs and Interconnection Networks: Diameter and vulnerability. Britain: Cambridge University Press, 003. [] Buckley, F. and Harary, F. Distance in Graphs, Perseus Books, [3] Buckley, F. and Lewinter, M. Graphs with all diametral paths through distant central vertices, Mathematical and Computer Modelling, 17, 1993, pp [4] Deogun, J. S. and Kratsch, D. Diametral Path Graphs, Graph Theoretic Concepts in Computer Science, 1017, 1995, pp [5] Harary, F. Graph Theory, New Delhi: Narosa Publishing House, 001. [6] Kaveh, A. Structural Mechanics: Graph and Matrix Methods, Tehran: Research Studies Press, 004. [7] Ore, O. Diameters in Graphs, Journal of Combinatorial Theory, 5, 1968, pp [8] Thomas, J. and Varghese, J. Decomposition in Total Graphs, Advanced Modelling and Optimization, 15(1), 013, pp editor@iaeme.com

8 Diametral Paths in Total Graphs of Complete Graphs, Complete Bipartite Graphs and Wheels [9] General Construction in Steel Code of Practice, Indian Standard 800, 007. [10] Plain and Reinforced Concrete - Code of Practice, Indian Standard 456, 000. [11] B. Sateesh Kumar, Dr. A.Govardhan Analysis of Bipartite Rankboost Approach for Score Level Fusion of Face and Palmprint Biometrics. International Journal of Computer Engineering and Technology (IJCET). 6(4), 015, pp [1] Uma Maheswari S. and Maheswari B. Independent Domination Number of Euler Totient Cayley Graphs and Arithmetic Graphs. International Journal of Advanced Research in Engineering and Technology, 7 (3), 016, pp editor@iaeme.com