1-Homogeneous Graphs with Cocktail Party, mu -Graphs by Jurisic A.

By Jurisic A.

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2-17, which is an Euler graph. Suppose that we start from vertex a and trace the path a b c. Fig. 2-17 Arbitrarily traceable graph from c. Now at c we have the choice of going to a, d, or e. If we took the first choice, we would only trace the circuit a b c a, which is not an Euler line. Thus, starting from a, we cannot trace the entire Euler line simply by moving along any edge that has not already been traversed. This raises the following interesting question: What property must a vertex v in an Euler graph have such that an Euler line is always obtained when one follows any walk from vertex v according to the single rule that whenever one arrives at a vertex one shall select any edge (which has not been previously traversed)?

CONTENTS PREFACE 1INTRODUCTION 1-1What is a Graph? 1-2Application of Graphs 1-3Finite and Infinite Graphs 1-4Incidence and Degree 1-5Isolated Vertex, Pendant Vertex, and Null Graph 1-6Brief History of Graph Theory Summary References Problems 2PATHS AND CIRCUITS 2-1Isomorphism 2-2Subgraphs 2-3A Puzzle With Multicolored Cubes 2-4Walks, Paths, and Circuits 2-5Connected Graphs, Disconnected Graphs, and Components 2-6Euler Graphs 2-7Operations On Graphs 2-8More on Euler Graphs 2-9Hamiltonian Paths and Circuits 2-10The Traveling Salesman Problem Summary References Problems 3TREES AND FUNDAMENTAL CIRCUITS 3-1Trees 3-2Some Properties of Trees 3-3Pendant Vertices in a Tree 3-4Distance and Centers in a Tree 3-5Rooted and Binary Trees 3-6On Counting Trees 3-7Spanning Trees 3-8Fundamental Circuits 3-9Finding All Spanning Trees of a Graph 3-10Spanning Trees in a Weighted Graph Summary References Problems 4CUT-SETS AND CUT-VERTICES 4-1Cut-Sets 4-2Some Properties of a Cut-Set 4-3All Cut-Sets in a Graph 4-4Fundamental Circuits and Cut-Sets 4-5Connectivity and Separability 4-6Network Flows 4-71-Isomorphism 4-82-Isomorphism Summary References Problems 5PLANAR AND DUAL GRAPHS 5-1Combinatorial Vs.

For instance, the two graphs shown in Fig. 2-4 satisfy all three conditions, yet they are not isomorphic. That the graphs in Figs. 2-4(a) and (b) are not isomorphic can be shown as follows: If the graph in Fig. 2-4(a) were to be isomorphic to the one in (b), vertex x must correspond to y, because there are no other vertices of degree three. Now in (b) there is only one pendant vertex, w, adjacent to y, while in (a) there are two pendant vertices, u and v, adjacent to x. Finding a simple and efficient criterion for detection of isomorphism is still actively pursued and is an important unsolved problem in graph theory.

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