Undirected graph definition - Math Insight
An undirected graph is graph, i.e., a set of objects (called vertices or nodes) that are connected together, where all the edges are bidirectional.
What is the definition of the density of a graph?
What about multigraphs? What would be the formula for the density of, respectively, an undirected and directed multigraph which has multiple edges and possibly loops? Thank you for the …
Closure of an Undirected Graph - Mathematics Stack Exchange
Aug 28, 2018 · Is the closure of a graph saying that we add edges between every vertex so they touch? I am not concerned about using this definition in particular, just a definition that …
Proving that the number of vertices of odd degree in any graph G …
Jan 23, 2015 · The sum of degrees of any graph can be worked out by adding the degree of each vertex in the graph. The sum of degrees is twice the number of edges. Therefore, the sum of …
graph theory - Show that there's a unique minimum spanning tree …
Apr 5, 2013 · Show that there's a unique minimum spanning tree (MST) in case the edges' weights are pairwise different $(w(e)\\neq w(f) \\text{ for } e\\neq f)$. I thought that the proof …
Proof a graph is bipartite if and only if it contains no odd cycles
Apr 29, 2024 · How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
Proof that the sum of all degrees is equal to twice the number of …
Aug 17, 2017 · $2n + 2 = 2 (n + 1) = \sum \limits_ {v \in V'} deg (v)$ Which proofs P (n + 1). Does the above proof make sense? I had a look at some other questions, but couldn't find a fully …
optimization - Show that a graph has a unique MST if all edges …
The question is about showing that the minimal spanning tree is unique if all the edges have different weights. If one goes through any of the greedy algorithms (Prim, Kruskal..) for finding …
Acyclic Undirected Graph - Mathematics Stack Exchange
Apr 29, 2015 · 1 HINT: Trees are simply the connected acyclic undirected graphs. Thus, every component of an acyclic undirected graph is a tree. (Indeed, another name for acyclic …
Laplace matrix for undirected graph - Mathematics Stack Exchange
Oct 12, 2023 · My point is how to calculate the Laplacian matrix for an undirected graph by using the incidence matrix. What is the incidence matrix for an undirected graph anyway?