Graph Theory

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

By Santanu Saha Ray

The e-book has many vital positive aspects which make it compatible for either undergraduate and postgraduate scholars in numerous branches of engineering and basic and technologies. the real subject matters interrelating arithmetic and desktop technology also are lined in brief. The e-book turns out to be useful to readers with a variety of backgrounds together with arithmetic, computing device Science/Computer purposes and Operational study. whereas facing theorems and algorithms, emphasis is laid on buildings which include formal proofs, examples with functions. Uptill, there's shortage of books within the open literature which disguise everything together with most significantly quite a few algorithms and functions with examples.

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Since, {b} is a cut set in T, {b} partitions all vertices of T into two disjoint sets. Consider the same partition of vertices in G and the cut set S in G that corresponds to this partition. Fig. 4 Fundamental Circuits and Fundamental Cut Sets 43 Cut set S will contain only one branch b of T, and the rest (if any) of the edges in S are chords with respect to T. Such a cut set S containing exactly one branch of a tree T is called a fundamental cut set with respect to T. 1 How many fundamental circuits and cut sets are there in a graph G with respect to any spanning tree with 10 vertices and 13 edges.

38 4 Trees and Fundamental Circuits Proof Let T be a tree; A, B be an arbitrary pair of vertices. Since, T is a connected graph so A and B are connected by a path. Let if possible, A and B be connected by two distinct paths. These two paths together form a circuit and then T cannot be a tree. So, there is only one path connecting A and B. 1) If there is one and only one path between every pair of vertices in a graph G, then G is a tree. Proof Existence of a path between every pair of vertices assures that G is connected.

Let, if possible, G be disconnected. Then G has two or more components. Without loss of generality, suppose G1 and G2 be two such components. Since, G1 and G2 are subgraphs of G they also do not contain any circuit. Let vj and vk be two vertices in the components G1 and G2 , respectively. Add an edge e between vj and vk (Fig. 5). 40 4 Trees and Fundamental Circuits Fig. 5 A connected graph G + e Since there is no path between vj and vk in G, so adding e would not create a circuit. , a tree. We see this tree is having n number of vertices and ðn À 1Þ þ 1 ¼ n number of edges.

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