Let’s walk through typical problems from Narsingh Deo’s Graph Theory and how a good solution approach looks.
Graph theory is visual. For any problem involving isomorphism or planarity, redraw the graph. Often, the solution reveals itself when you see the dual graph or the bridge structure. Graph Theory By Narsingh Deo Exercise Solution
Problem: Show that the sum of the degrees of all vertices in a finite undirected graph is twice the number of edges. Let’s walk through typical problems from Narsingh Deo’s
Solution: Proof: Let $G = (V, E)$ be a graph with $n$ vertices and $e$ edges. Every edge in a graph connects two vertices (or a vertex to itself in a loop). Therefore, every edge contributes 2 to the total sum of degrees. Therefore: $$ \sum_i=1^n deg(v_i) = 2 \times |E| $$
Therefore: $$ \sum_i=1^n deg(v_i) = 2 \times |E| $$
Topic: Graph Terminology, Degrees, and Types of Graphs.