# Graph Theory with Algorithms and its Applications: In by Santanu Saha Ray

By Santanu Saha Ray

The ebook has many very important positive factors which make it appropriate for either undergraduate and postgraduate scholars in a number of branches of engineering and basic and technologies. the $64000 subject matters interrelating arithmetic & desktop technology also are coated in brief. The e-book comes in handy to readers with quite a lot of backgrounds together with arithmetic, laptop Science/Computer functions and Operational study. whereas facing theorems and algorithms, emphasis is laid on structures which encompass formal proofs, examples with purposes. Uptill, there's shortage of books within the open literature which disguise all the pieces together with most significantly numerous algorithms and functions with examples.

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**Sample text**

14 3. Show that the following graphs are isomorphic and each graph has the same bipartition (Fig. 15). Fig. 15 4. What is the difference between a closed trail and a cycle? 5. Are the following graphs isomorphic? (Fig. 16). 5 Cycles 23 Fig. 16 6. Prove that a simple graph having n number of vertices must be connected if it has more than ðn À 1Þðn À 2Þ=2 edges. 7. Check whether the following two given graphs G1 and G2 are isomorphic or not (Fig. 17). Fig. 17 8. Prove that the number of edges in a bipartite graph with n vertices is at mostðn2 =2Þ.

Let v 2 G be an arbitrary chosen vertex. , Y is the set of all vertices y of G with the property that any shortest v À y path of G has odd length. Here, dG ðu; vÞ ¼ shortest distance from the vertex u to the vertex v n o k ¼ min k : u ! , dG ðu; vÞ\1 for 8u; v 2 G. Otherwise, G is disconnected] Then clearly, since the graph G is connected V ¼ X [ Y and also by definition of distance X \ Y ¼ ;. Now, we show that V ¼ X [ Y is a bipartition of G by showing that any edge of G must have one end vertex in X and another in Y: Suppose that u; w 2 V ðGÞ are both either in X or in Y and they are adjacent.

This is again a contradiction, since the longest path in T has length m. Thus, dðv0 Þ ¼ 1: Similarly, we can show that dðvm Þ ¼ 1: So, we see that v0 and vm are pendant and they are distinct. This completes the proof. 5 A tree with n number of vertices has n À 1 number of edges. Proof Let, T be a tree. The theorem will be proved by method of induction on n. Clearly, the result is true for n = 1, 2. We assume that, the result is true for k number of vertices whenever k\n. In T (Fig. 4), let e be an edge with end vertices A and B.