Show that isomorphism of simple graphs is an equivalence relation.

Prove that isomorphism is an equivalence relation on the class of all rings.

Define isomorphism of directed graphs.

a) What does it mean for two simple graphs to be isomorphic? b) What is meant by an invariant with respect to isomorphism for simple graphs? Give at least five examples of such invariants. c) Give an example of two graphs that have the same numbers of vertices, edges, and degrees of vertices, but that are not isomorphic. d) Is a set of invariants known that can be used to efficiently determine whether two simple graphs are isomorphic?

What does it mean for two graphs to be the same? Let G and H be graphs. We say that G is isomorphic to H provided that there is a bijection f:V(G)$\rightarrow$V(H) such that for all a,b$\in$V(G) we have a$\sim$b (in G) if and only if f(a)$\sim$f(b) (in H). The function f is called an isomorphism of G to H. We can think of f as renaming the vertices of G with the names of the vertices in H in a way that preserves adjacency. Less formally, isomorphic graphs have the same drawing (except for the names of the vertices). Please do the following: a. Prove that isomorphic graphs have the same number of vertices. b. Prove that if f:V(G)$\rightarrow$V(H) is an isomorphism of graphs G and H and if vV(G), then the degree of v in G equals the degree of f(v) in H. c. Prove that isomorphic graphs have the same number of edges. d. Give an example of two graphs that have the same number of vertices and the same number of edges but that are not isomorphic. e. Let G be the graph whose vertex set is {1,2,3,4,5,6}. In this graph, there is an edge from v to w if and only if v-w is odd. Let H be the graph in the figure. Find an isomorphism f:V(G)$\rightarrow$V(H). Figure: vertices a,b,c,d,e,f where a,b,c are connected to d,e,f each.

Show that if G is a self-complementary simple graph with v vertices, then v ≡ 0 or 1 (mod 4).

Construct a complete binary tree of height 4 and a complete 3-ary tree of height 3.

$$\begin{array} { l } { \text { Prove that isomorphism is an equivalence relation. That is, for any } } \\ { \text { groups } G , H , \text { and } K , G \approx G , G \approx H \text { implies } H \approx G , \text { and } G \approx H \text { and } } \\ { H \approx K \text { implies } G \approx K } \end{array}$$

What is the sum of the entries in a row of the incidence matrix for an undirected graph?

Suppose that a connected bipartite planar simple graph has e edges and v vertices. Show that e ≤ 2v − 4 if v ≥ 3.

Show that if G and H are isomorphic directed graphs, then the converses of G and H are also isomorphic.

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Someone in your class can speak Hindi. b) Everyone in your class is friendly. c) There is a person in your class who was not born in California. d) A student in your class has been in a movie. e) No student in your class has taken a course in logic programming.

A simple graph G is self-complementary if G and $\bar{G}$ are isomorphic. (a) Find a self-complementary graph having five vertices. (b) Find another self-complementary graph.

Recall the definition of graph isomorphism from Exercise 47.21. We call a graph $G$ self-complementary if $G$ is isomorphic to $\overline{G}$. a. Show that the graph $G=(\{a,b,c,d\},\{ab,bc,cd\})$ is self-complementary. b. Find a self-complementary graph with five vertices. c. Prove that if a self-complementary graph has n vertices, then n$\equiv$0 (mod 4) or n$\equiv$1 (mod 4).