ch3 - CHAPTER 3 Diameters and eigenvalues 3.1 The diameter...

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CHAPTER 3 Diameters and eigenvalues 3.1. The diameter of a graph In a graph G , the distance between two vertices u and v , denoted by d ( u, v ), is defined to be the length of a shortest path joining u and v in G . (It is possible to define the distance by various more general measures.) The diameter of G , denoted by D ( G ), is the maximum distance over all pairs of vertices in G .T h e diameter is one of the key invariants in a graph which is not only of theoretical interest but also has a wide range of applications. When graphs are used as models for communication networks, the diameter corresponds to the delays in passing messages through the network, and therefore plays an important role in performance analysis and cost optimization. Although the diameter is a combinatorial invariant, it is closely related to eigenvalues. This connection is based on the following simple observation: Let M denote an n × n matrix with rows and columns indexed by the vertices of G . Suppose G satisfies the property that M ( u, v )=0if u and v are not adjacent. Furthermore, suppose we can show that for some integer t , and some polynomial p t ( x ) of degree t ,wehave p t ( M )( u, v ) 6 =0 for all u and v . Then we can conclude that the diameter D ( G ) satisfies: D ( G ) t. Suppose we take M to be the sum of the adjacency matrix and the identity matrix and the polynomial p t ( x )tobe(1+ x ) t . The following inequality for regular graphs which are not complete graphs can then be derived (which will be proved in Section 3.2 as a corollary to Theorem 3.1; also see [ 59 ]): (3.1) D ( G ) ± log( n - 1) log(1 / (1 - λ )) ² . Here, λ basically only depends on λ 1 . For example, we can take λ = λ 1 if 1 - λ 1 λ n - 1 - 1. In general, we can slightly improve (3.1) by using the same “spectrum shifting” trick as in Section 1.5 (see Section 3.2). Namely, we define λ =2 λ 1 / ( λ n - 1 + λ 1 ) 2 λ 1 / (2 + λ 1 ) ,andwethenhave (3.2) D ( G ) & log( n - 1) log λ n - 1 + λ 1 λ n - 1 - λ 1 . 41
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42 3. DIAMETERS AND EIGENVALUES We note that for some graphs the above bound gives a pretty good upper bound for the diameter. For example, for k -regular Ramanujan graphs (defined later in 6.3.6), we have 1 - λ 1 = λ n - 1 - 1=1 / (2 k - 1) so we get D log( n - 1) / (2 log( k - 1)) which is within a factor of 2 of the best possible bound. The bound in (3.1) can be further improved by choosing p t to be the Chebyshev polynomial of degree t . We can then replace the logarithmic function by cosh - 1 (see [ 65 ] and Theorem 3.3): D ( G ) & cosh - 1 ( n - 1) cosh - 1 λ n - 1 + λ 1 λ n - 1 - λ 1 . The above inequalities can be generalized in several directions. Instead of con- sidering distances between two vertices, we can relate the eigenvalue λ 1 to distances between two subsets of vertices (see Section 3.2). Furthermore, for any k 1, we can relate the eigenvalue λ k to distances among k + 1 distinct subsets of vertices (see Section 3.3).
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This note was uploaded on 09/07/2011 for the course MATH 262 taught by Professor Aterras during the Spring '08 term at UCSD.

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ch3 - CHAPTER 3 Diameters and eigenvalues 3.1 The diameter...

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