By W.D. Wallis
Concisely written, mild advent to graph thought compatible as a textbook or for self-study
Graph-theoretic purposes from diversified fields (computer technological know-how, engineering, chemistry, administration science)
2nd ed. comprises new chapters on labeling and communications networks and small worlds, in addition to elevated beginner's material
Many extra alterations, advancements, and corrections caused by school room use
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Additional info for A Beginner’s Guide to Graph Theory
3 Prove that a finite graph on v vertices that contains no cycle is connected if and only if it has v - 1 edges. 4 Prove that a connected graph is a tree if and only if it has the following property: If x and y are distinct vertices, then there is a unique path in G from x to y. 1. 7. Prove that no tree other than Kt or Kz is a perfect square. 6 Give an example of an infinite "tree" that contains no vertex of degree 1. 1. 7 Let T be a tree on v vertices, v ::: 5, with precisely four vertices of degree 1 each and precisely one vertex of degree 4.
Moreover, a must be an edge in that cycle. Select an edge b of the cycle, other than a. Then T + a - b will be acyclic, and it is still connected, so it is a tree. In particular, suppose R is a spanning tree of G that has k edges in common with t, and suppose a is an edge of R (but not ofT). The cycle in T + a must contain an edge that is not in R, because otherwise R would contain a cycle. If such an edge is chosen as b, then the tree T - a + b will have k + 1 edges in common with R. Call this tree Tt.
Go along this edge to its other endpoint, say y. Then choose any other edge incident with y. In general, on arriving at a vertex, select any edge incident with it that has not yet been used, and go along the edge to its other endpoint. At the moment when this walk has led into the vertex z, where z is not x, an odd nurober of edges touching z has been used up ( the last edge to be followed, and an even number previously). Since z is even, there is at least one edge incident with z that is still available.
A Beginner’s Guide to Graph Theory by W.D. Wallis