Why a complete graph has $\frac {n (n-1)} {2}$ edges?
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 …
Proving that the number of vertices of odd degree in any graph G …
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 …
How many nonisomorphic directed simple graphs are there with …
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 …
combinatorics - Deleting vertices of odd degree from a graph ...
Nov 11, 2024 · To see this, consider a complete graph with $5$ vertices, to which we adjoin $v_1$ and $v_2$ (each is connected to one node of the complete part of the graph and they …
Prove that every triangle-free graph on $n$ vertices has chromatic ...
Dec 1, 2024 · The easiest way to do this is with the probabilistic method. I'll give you a quick outline: Prove that every triangle-free graph has at most $\lfloor n^2 /4\rfloor$ edges. …
Proof verification: Prove that a tree with n vertices has n-1 edges
Mar 20, 2015 · The proof is incorrect/incomlete. What you are fundamentally saying is that if you have a tree with n vertex and n-1 edges, you can obtain a tree with n+1 vertices and n edges. …
How to find non-isomorphic trees? - Mathematics Stack Exchange
"Draw all non-isomorphic trees with 5 vertices." I have searched the web and found many examples of the non-isomorphic trees with 5 vertices, but I can't figure out how they have …
Relationship between vertices and edges in platonic solids
Apr 4, 2025 · The Question: Given a list of the vertices in a platonic solid, is there a way to calculate which vertices are connected by an edge? I know that one could find edges using …
Why does a 3-regular planar graph of diameter 3 have at most 12 …
Jun 1, 2015 · Today, I saw an interesting exercise on page 224 of the West textbook "Introduction to Graph Theory". 6.1.15. Construct a 3-regular planar graph of diameter 3 with 12 vertices. …
How to sort vertices of a polygon in counter clockwise order?
16 How to sort vertices of a polygon in counter clockwise order? I want to create a function (algorithm) which compares two vectors $\vec v$ and $\vec u$ which are vertices in a polygon. …