   Chapter 10.4, Problem 31ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
459 views

# a. Prove that the following is an invariant for graph isomorphism: A vertex of degree i is adjacent to a vertex of degree j. b. Find all nonisomorphic trees with six vertices.

To determine

(a)

To prove:

A vertex of degree i is adjacent to a vertex of degree jis an invariant for graph isomorphism.

Explanation

Given information:

A vertex of degree i is adjacent to a vertex of degree j.

Proof:

Let G and G be isomorphic graphs and let the graph G contain adjacent vertices v and w such that vertex v and w such that vertex v has degree i and such that vertex w has degree j (for some nonnegative integers i, j ).

We then need to show that G has aa vertex of degree i that adjacent to a vertex of degree j.

Let V(G) be the set of vertices of G and let V(G) be the set of vertices of G.

Let E(G) be the set of edges of G and let E(G) be the set of edges of G.

Since G and G are isomorphic, there exists two one-to-one correspondences

g:V(G)V(G') and h:E(G)E(G') such that

v is an endpoint of eg(v) is an endpoint of h(e).

Since v has degree i, there are i distinct edges e1,e2,...,ei in E(G) that have v as its endpoint.

g(v) is then also an endpoint of h(e1),h(e2),...,h(ei).

Moreover, h(e1),h(e2),...,h(ei) are distinct edges, because e1,e2,

To determine

(b)

To find:

All nonisomorphic trees with six vertices.

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