1 An edge "e" in a graph (Undirected or directed ) that is associated with the pair of vertices n and q is said to be incident on n and q, and n and q are said to be incident on e and to be adjacent …

Oct 14, 2023 · Now, let's use these properties to prove the statement. If a graph has an Eulerian path, there must be exactly two vertices with odd degrees (the starting and ending vertices) …

Jan 23, 2015 · To prove that the number of odd vertices in a simple graph is always even, we can use the Handshaking Lemma, which states that the sum of the degrees of all vertices in a …

4 $\frac {n (n-1)} {2} = \binom {n} {2}$ is the number of ways to choose 2 unordered items from n distinct items. In your case, you actually want to count how many unordered pair of vertices …

A directed simple graph is a structure consisting of the set of vertices and a binary relation that is irreflexive. For the case of the disconnected graph, the relation is empty, and there is one such …

Mar 10, 2016 · In case 1, any choice of three distinct vertices will form a triangle, so case 1 contributes $\binom {5} {3}=10$ to the overall sum. In case 2, notice that any choice of two …

Dec 11, 2010 · Anyone know of an online tool available for making graphs (as in graph theory - consisting of edges and vertices)? I have about 36 vertices and even more edges that I wish to …

Dec 6, 2017 · I understand the theory, but some of the questions require memorizing several formulas for calculating the number of vertices, internal vertices, and leaves. The professor …

Here's alternative proof that a connected graph with n vertices and n-1 edges must be a tree modified from yours but without having to rely on the first derivation:

I have tried some ways - mainly using induction by removing one of the vertices of the set from the graph, and/or using Menger's theorem to construct the cycle. But I always encounter …