📘 graph theory

1. **Problem Statement:** Determine if the given graph with vertices A, B, C, D, E, F, G, H, I, J, K and specified edges has an Eulerian cycle or an Eulerian path. 2. **Definitions

1. **Problem Statement:** We are given a graph with cities A, B, C, D, E, X connected by flight routes with given weights. We need to determine which of the given flight paths is a

1. **State the problem:** We are given a graph with vertices A, B, C, D, E, X and edges representing flight routes with weights as airfare costs. We need to find the total number o

1. **Prove that Complete Graph K₄ is Planar** A graph is planar if it can be drawn on a plane without any edges crossing.

1. **Prove that Complete Graph K₄ is Planar** The problem is to prove that the complete graph on 4 vertices, $K_4$, can be drawn on a plane without any edges crossing.

1. Graph theory is a branch of mathematics that studies graphs, which are mathematical structures used to model pairwise relations between objects. 2. A graph consists of vertices

1. The problem describes a rectangular grid of nodes connected by edges forming an H-like shape with nodes arranged in two vertical columns of three nodes each and a horizontal row

1. The problem is to find the diameter of a generalized Petersen graph. 2. A generalized Petersen graph $GP(n,k)$ is defined for integers $n \geq 3$ and $1 \leq k < \frac{n}{2}$, w

1. **Stating the problem:** We are given two graphs with vertices and edges described, and we need to determine if these two graphs are isomorphic. 2. **Definition of graph isomorp

1. **Problem Statement:** We need to find the maximum height of a truck that can travel from Start to Finish along the roads shown, where each road has a height limit. 2. **Underst

1. **Problem Statement:** Find the length of the shortest path from vertex $a$ to vertex $z$ in the given weighted graph using Dijkstra's algorithm. 2. **Dijkstra's Algorithm Overv

1. The problem asks to identify the correct property of a complete graph. 2. A complete graph is a graph in which every pair of distinct vertices is connected by a unique edge.

1. The problem is to understand the properties of a tree in graph theory. 2. A tree is defined as a connected graph with no circuits (cycles).

1. The problem is to find a spanning tree from the given grid of quadrilaterals and calculate the total length of the spanning tree edges. 2. A spanning tree connects all vertices

1. **Problem Statement:** State and prove the Handshaking theorem and use it to show that if all vertices of an undirected graph have degree $k$, then the number of edges is a mult

1. **Problem Statement:** State and prove the Handshaking Theorem and use it to show that if all vertices of an undirected graph have degree $k$, then the number of edges is a mult

1. **Énoncé du problème** : Appliquer l'algorithme de Bellman-Ford pour trouver les plus courts chemins depuis un sommet source donné dans le graphe orienté avec poids donnés. 2. *

1. **Énoncé du problème** : Appliquer l'algorithme de Bellman-Ford sur le graphe donné pour trouver les plus courts chemins à partir d'un sommet source. 2. **Description du graphe*

1. The problem asks for an example of a simple graph with 12 vertices and 35 edges. 2. A simple graph is an undirected graph with no loops or multiple edges between the same pair o

1. **State the problem:** We are given that $G$ is a simple loopless Eulerian graph, $H$ is an Eulerian subgraph of $G$, and we want to prove that the graph $G \setminus H$ (the gr

1. Problem: Find the in-degree and out-degree of each vertex in the given directed graphs. ---