Graphs and Matrices (Universitext)

Graphs and Matrices (Universitext)

Language: English

Pages: 193

ISBN: 1447165683

Format: PDF / Kindle (mobi) / ePub


This new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices associated with graphs (for example, incidence, adjacency and Laplacian matrices) are treated in detail.

Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph.

Such an extensive coverage of the subject area provides a welcome prompt for further exploration. The inclusion of exercises enables practical learning throughout the book.

In the new edition, a new chapter is added on the line graph of a tree, while some results in Chapter 6 on Perron-Frobenius theory are reorganized.

Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory, it will also benefit readers in the sciences and engineering.

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Theory L.W. Beineke and R.J. Wilson, Ed. Academic Press, New York, pp. 337–360 (1978). [God93] Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall Inc, New York (1993) [KM01] Koolen, J.H., Moulton, V.: Maximal energy graphs. Adv. Appl. Math. 26, 47–52 (2001) Chapter 7 Line Graph of a Tree Let G be a graph with V (G) = {1, . . . , n} and let A(G) (or simply, A) denote the adjacency matrix of G. If Δ is the diagonal matrix of vertex degrees, then recall that L = Δ − A is the Laplacian

is a tree and C1 = 0. It follows by Lemma 8.10 that rank(C) = n − 1. Then rank(L − μI ) is n − 1 as well, and hence μ has algebraic multiplicity 1. Let T be a tree with V (T ) = {1, . . . , n}. Let L be the Laplacian of T and μ the algebraic connectivity. Let x be a Fiedler vector and suppose x has no zero coordinate. Then by Theorem 8.11, μ has algebraic multiplicity 1, and hence any other Fiedler vector must be a scalar multiple of x. Thus, in this case there is an edge of T that is the

vector. Then one of the following cases occur. (i) No entry of x is zero. In this case there is a unique edge e = {i, j} such that xi > 0 and x j < 0. Further, along any path in T that starts at i and does not contain j, the entries of x increase, while along any path in T that starts at j and does not contain i, the entries of x decrease. (ii) Some entry of x is zero. In this case the subgraph of T induced by the zero vertices is connected. There is a unique vertex k such that xk = 0 and k is

H and the p) . Thus, the sum of the degrees (in H ) of the vertices not in H is at least μ p(n− n vertices in H is at most kp − μ p(n − p) pμ =p +k−μ . n n Hence, the average degree of a vertex in H is at most pμ n + k − μ. Let G be a connected graph with V (G) = {1, . . . , n}. Let V1 be a nonempty subset of V (G) and let b(V1 ) be the number of edges with precisely one endpoint n 1) in V1 . The minimum value of b(V |V1 | taken over all V1 with |V1 | ≤ 2 is called the isoperimetric number

ij-path P of length d(i, j). Orient each edge in P in the direction from i to j and assign an arbitrary orientation to the remaining edges of G. If g : E(G) → IR is defined as g(ek ) = 1, if ek ∈ P 0, otherwise then g is a unit flow from i to j. Since r(i, j) is the minimum value of the squared norm of a unit flow from i to j, we have d(i, j) = ||g||2 ≥ r(i, j), and the proof is complete. It can be shown that equality holds in (10.5) if and only if there is a unique ij-path. Before proving this

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