📘 graph theory

1. **Problem Statement:** Determine which of the given graphs are Euler graphs. 2. **Definition:** An Euler graph is a graph containing an Eulerian circuit, which is a cycle that u

1. **Problem Statement:** Find the number of vertices, number of edges, degree of each vertex, and identify isolated and pendant vertices for the graphs G_1, G_2, and G_3. 2. **Def

1. **Problem Statement:** Prove that there is no simple graph with five vertices having degrees 4, 4, 4, 2, 2.

1. **Problem Statement:** Prove that there is no simple graph with five vertices having degrees 4,4,4,2,2.

1. **Problem Statement:** Construct the intersection graph of the collection of sets $A_1, A_2, A_3, A_4, A_5$ where each vertex represents a set and edges connect vertices whose s

1. **Problem Statement:** Determine if the given graph with vertices A, B, C, D, F and edges as described is Eulerian. If yes, find the Euler circuit; if no, explain why. 2. **Eule

1. **Problem Statement:** Find the minimum spanning tree (MST) of the given weighted graph using Prim's algorithm.

1. **Problem:** Determine the properties of the given graphs regarding directed/undirected edges, multiple edges, and loops. 2. **Explanation:**

1. The problem asks about a Hamiltonian path that ends in the same place it began. 2. A Hamiltonian path is a path in a graph that visits each vertex exactly once.

1. The problem asks about the implication on a connected graph if the number of vertices with odd degree is 1. 2. According to graph theory, specifically Eulerian path and circuit

1. The problem asks to find the value of $\sum d(v)$, which is the sum of the degrees of all vertices in the graph. 2. The degree $d(v)$ of a vertex $v$ is the number of edges inci

1. **Problem Statement:** Construct a graph $G$ with 14 vertices such that: - Minimum degree $\delta(G) = 3$

1. **Problem Statement:** We are given a directed graph with nodes P, Q, R, S and edges: P\to Q, Q\to Q (loop), Q\to R, R\to P, R\to S, and S\to R. We need to analyze or solve a pr

1. **Problem Statement:** Determine which of the given sequences are circuits in the graph with vertices $V_1, V_2, V_3, V_4, V_5$ and edges $a(V_1\text{-}V_2), b(V_2\text{-}V_3),

1. **Problem Statement:** Find the shortest eligible walk in the given graph with vertices V1, V2, V3, V4 and edges e1 (V1-V2), e2 (V1-V3), e3 (V1-V4), e4 (V3-V4). The shortest wal

1. **Énoncé du problème :** Nous avons deux graphes non orientés G et H avec leurs matrices d'adjacence respectives $M_G$ et $M_H$.

1. Problem: Find the end vertices of edge e8 in the given graph. 2. The graph description states that edge e8 connects vertices V2 and V5.

1. Problem: Find the end vertices of edge e8 in the given graph. 2. Given: Edge e8 connects vertex V2 and vertex V5.

1. **Problem Statement:** Determine if the given graph has (i) an Euler's path and (ii) an Euler's circuit. 2. **Recall Definitions:**

1. **Problem Statement:** Construct the Minimum Spanning Tree (MST) of the given graph using Prim's algorithm and find the total cost. 2. **Prim's Algorithm Overview:**

1. **Problem Statement:** Find the Minimum Spanning Tree (MST) of the given graph using Prim's algorithm. 2. **Graph Details:**