By Adrian Bondy, U. S. R. Murty

The first goal of this booklet is to provide a coherent advent to graph concept, appropriate as a textbook for complicated undergraduate and starting graduate scholars in arithmetic and desktop technology. It offers a scientific therapy of the speculation of graphs with out sacrificing its intuitive and aesthetic charm. normal evidence innovations are defined and illustrated. The booklet additionally serves as an advent to analyze in graph theory.

This publication is a heavily-revised model of the sooner Graph concept With purposes.

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Decompositions . . . . . . . . . . . . . . . . . . . Proof Technique: Linear Independence . . . . . . . Coverings . . . . . . . . . . . . . . . . . . . . . . 5 Edge Cuts and Bonds . . . . . . . . . . . . . . . Edge Cuts . . . . . . . . . . . . . . . . . . . . . . Bonds . . . . . . . . . . . . . . . . . . . . . . . . Cuts in Directed Graphs . . . . . . . . . . . . .

For historical reasons, this graph is known as the line graph of G and denoted L(G). 17 depicts a graph and its line graph. 1 12 2 4 24 3 G 23 34 L(G) Fig. 17. 2). As for the Fano plane, its intersection graph is isomorphic to K7 , because any two of its seven lines have a point in common. The deﬁnition of the line graph L(G) may be extended to all loopless graphs G as being the graph with vertex set E in which two vertices are joined by just as many edges as their number of common ends in G. When V = R and F is a set of closed intervals of R, the intersection graph of (V, F ) is called an interval graph.

When dealing with problems of traﬃc ﬂow, for example, it is necessary to know which roads in the network are one-way, and in which direction traﬃc is permitted. Clearly, a graph of the network is not of much use in such a situation. What we need is a graph in which each link has an assigned orientation, namely a directed graph. Formally, a directed graph D is an ordered pair (V (D), A(D)) consisting of a set V := V (D) of vertices and a set A := A(D), disjoint from V (D), of arcs, together with an incidence function ψD that associates with each arc of D an ordered pair of (not necessarily distinct) vertices of D.

### Graph Theory (Graduate Texts in Mathematics, Volume 244) by Adrian Bondy, U. S. R. Murty

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