Basic Graph Terminologies

Abstract

In this chapter, we learn some definitions of basic graph theoretic terminologies and know some preliminary results of graph theory.

Exercises

• 1. Show that every regular graph with an odd degree has an even number of vertices.

• 2. Construct the complement of $K_{3,3}$, $W_5$, and $C_5$.

• 3. Can you construct a disconnected graph G of two or more vertices such that $\overline{G}$ is also disconnected. Give a proof supporting your answer.

• 4. Give two examples of self-complementary graphs.

• 5. What is the necessary and sufficient condition for $K_{m,n}$ to be a regular graph?

• 6. Is there a simple graph of n vertices such that the vertices all have distinct degrees? Give a proof supporting your answer.

• 7. Draw the graph $G=(V,E)$ with vertex set $V=\{a,b,c,d,e,f,g,h\}$ and edge set $\{(a,b), (a,e), (b,c), (b,d), (c,d), (c,g), (d,e) (e,f), (f,g), (f,h), (g,h)\}$. Draw $G-(d,e)$. Draw the subgraph of G induced by $\{c,d,e,f\}$. Contract the edge (d, e) from G.

• 8. Show that two graphs are isomorphic if and only if their complements are isomorphic.