📘 graph theory

1. **Problem statement:** Given a $k$-regular graph $G = (V, E)$ where $k \in \mathbb{N}$, prove or disprove that the number of edges $|E|$ is always a multiple of $k$ for even and

1. **Problem statement:** We have a directed graph $D = (V, A)$ with nodes $V = \{1,2,3,4,5,6,7\}$ and directed edges $A = \{(1,2),(2,4),(3,1),(3,2),(3,4),(4,5),(5,2),(5,6),(6,3),(

1. **Problem statement:** Make all possible different connected simple graphs of order 4.

1. **Problem Statement:** Find the diameter of each graph type $N_n$, $K_n$, $P_n$, $C_n$, and $W_n$ and explain what $N$, $C$, $P$, and $W$ stand for. 2. **Definitions of Graphs:*

1. **Determine which graphs of these families are the same:** - $N_1 = K_1$ because $N_1$ is a null graph with 1 vertex and no edges, which is the same as the complete graph $K_1$.

1. The **order** of a graph is defined as the number of vertices (nodes) it contains. 2. The **size** of a graph is defined as the number of edges (connections) it contains.

1. The problem asks to find all simple connected graphs where the distance between every two non-adjacent vertices is exactly 2. 2. Let's analyze the condition: for any two vertice

1. **State the problem:** Determine if the sequence $5, 4, 3, 2, 1, 0$ is graphic, meaning it can represent the degree sequence of a simple graph. 2. **Recall the definition:** A s

1. Problem 1 asks to prove the equivalence of three statements about a graph $G$ with $v-1$ edges: (a) $G$ is connected, (b) $G$ is acyclic, and (c) $G$ is a tree. 2. Recall the de

1. **Problem Statement:** Given an adjacency matrix of a graph, our tasks are: a) Verify the given parts about the graph.

1. Prove that G and H are isomorphic. Both graphs have 6 vertices.

1. The problem: State and prove Konig's theorem, which relates the maximum matching size and minimum vertex cover size in bipartite graphs. 2. Statement of Konig's theorem: In any

1. **State the problem:** We have buses traveling between three Dzongkhags: Thimphu, Paro, and Wangdue. We need to first create a directed graph (digraph) showing bus routes (part

1. **Stating the problem:** We are given a table that shows the number of buses travelling between three Dzongkhags: Thimphu, Paro, and Wangdue. (i) We need to create a digraph rep

1. Masalah: Anda meminta lukisan bentuk rangkaian graf. 2. Penjelasan: "Rangkaian graf" biasanya merujuk pada representasi visual dari graf, yang terdiri dari simpul (nodes) dan si

1. **Nyatakan Masalah / State the Problem** Kita diberi jadual yang menunjukkan jenis buah-buahan kegemaran beberapa pelajar dan diminta:

1. The description you provided is about a graph of nodes with arrows connecting them, but no explicit problem or numerical values are given. 2. "Solve connect with dots" is unclea

1. **Problem 1: Given graph G with vertices and edges, find vertex set, edge set, degree of vertices, order, and size of G.** Step 1: Identify the vertex set $V$ from the problem s

1. The problem is to illustrate a flow chart from the given vertices and edges. 2. The vertices are: $A, B, C, D, E, F, G, H$.

1. **Given the graph $G$ with vertices $v_1$ to $v_8$ and specified edges:** - **Vertex set $V$:** The set of all vertices is

1. **Problem Statement:** We need to find a nontrivial connected graph $G$ which satisfies the following properties: (a) Every bridge of $G$ is adjacent to an edge that is not a br