By Sergei Ovchinnikov

ISBN-10: 1461407966

ISBN-13: 9781461407966

ISBN-10: 1461407974

ISBN-13: 9781461407973

This introductory textual content in graph concept makes a speciality of partial cubes, that are graphs which are isometrically embeddable into hypercubes of an arbitrary size, in addition to bipartite graphs, and cubical graphs. This department of graph conception has built speedily in the past 3 a long time, generating interesting effects and developing hyperlinks to different branches of arithmetic.

Currently, *Graphs and Cubes* is the single e-book on the market that offers a accomplished insurance of cubical graph and partial dice theories. Many workouts, in addition to old notes, are incorporated on the finish of each bankruptcy, and readers are inspired to discover the routines absolutely, and use them as a foundation for learn tasks.

The must haves for this article comprise familiarity with easy mathematical recommendations and strategies at the point of undergraduate classes in discrete arithmetic, linear algebra, staff idea, and topology of Euclidean areas. whereas the ebook is meant for lower-division graduate scholars in arithmetic, it is going to be of curiosity to a much broader viewers; due to their wealthy structural homes, partial cubes look in theoretical computing device technology, coding idea, genetics, or even the political and social sciences.

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**Extra info for Graphs and Cubes**

**Example text**

B) |V | = |E|. c) there is an edge e ∈ E such that G − e is a tree. 28. Prove that a graph is a tree if and only if it has exactly one spanning tree. 29. The center of a connected graph G = (V, E) is the set of vertices u such that maxv∈V d(u, v) is as small as possible. a) Let T be a tree on more than two vertices, and let T be a tree obtained from T by deleting all its leaves. Show that T and T have the same centers. b) Show that the center of a tree consists of a single vertex or the ends of an edge.

5). The length of any xv-walk is greater than or equal to d(x, v), therefore the xv-walk obtained by concatenating the edge xy, a shortest yupath, and the edge uv is a shortest xv-path containing edges e and f. 3) are of considerable interest later in the book. 13. Let e = xy and f = uv be two edges of a connected graph G = (V, E). The edge e is in relation Θ to the edge f if d(x, u) + d(y, v) = d(x, v) + d(y, u). 4) More formally, Θ = {(xy, uv) ∈ E × E : d(x, u) + d(y, v) = d(x, v) + d(y, u)} The relation Θ is a binary relation on the edge set E.

1). A walk in a bipartite graph G[X, Y ] is even if and only if its ends belong to the same part of the bipartition (X, Y ). Equivalently, a walk is odd if and only if its ends belong to different parts of (X, Y ). 1. 1. Even and odd paths in the graph K3,3 . It follows that a closed walk in a bipartite graph must be of even length. This gives us a necessary condition for a graph to be bipartite. In fact this condition is also sufficient. 1. A graph G = (V, E) is bipartite if and only if it contains no closed walk of odd length.

### Graphs and Cubes by Sergei Ovchinnikov

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