Download PDF by Xu B.G.: A 3-color Theorem on Plane Graphs without 5-circuits

By Xu B.G.

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In this book, only finite graphs are needed, which are referred to as graphs. The above definitions correspond to abstract graphs; however, a graph may be visualised as a set of points connected by line segments in Euclidean space; the nodes of a graph are identified with points, and its members are identified as line segments without their end points. Such a configuration is known as a topological graph. These definitions are illustrated in Fig. 16. 20 1 Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs Fig.

Therefore the cutset vectors of a graph form a vector space, the dimension of which is given by: rank ðSÞ ¼ ρðSÞ ¼ NðSÞ À b0 ðSÞ: ð1:44Þ As an example, the rank of S in Fig. 16a is ρ(S) ¼ 6 À 1 ¼ 5. 3 Orthogonality Property Two vectors are called orthogonal if their scalar product is zero. It can be shown that a vector is a cycle set (cutset) vector, if and only if it is orthogonal to every vector of a cutset (cycle set) basis. 5 Vector Spaces Associated with a Graph 27 vectors which represent subsets of the members, therefore the cycle set and cutset spaces are orthogonal components of each other.

This matrix is symmetric, and the row sums of A are the degrees of the nodes of S. The adjacency matrix of the graph S, shown in Fig. 28, is a 5 Â 5 matrix as: It can be noted that A is a symmetric matrix of trace zero. The (i, j)th entry of A2 shows the number of walks of length 2 with ni and nj as end nodes. Similarly, the entry in the (i, j) position of Ak is equal to the number of walks of length k with ni and nj as end nodes. The polynomial, f ðλÞ ¼ det ðIλ À AÞ, ð1:49Þ is called the characteristic polynomial of S.

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A 3-color Theorem on Plane Graphs without 5-circuits by Xu B.G.

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