Let k and n be positive integers. Let G be the graph with vertex set (n] such that u and e are adjacent if and only if k | u- v. Find, with explanation, the order and size of G.

Q: Prove that in any graph G with n vertices, n ≤ α(G)χ(G)

A: Consider a graph G on n vertices. Let the chromatic number χG=k. Then, k is the minimum number of…

Q: Prove that if G is a graph with no isolated vertices, then α'(G) +β'(G) = n(G)

A: Here α'(G) is largest size of matching in G. β'(G) is the smallest size of an edge cover. We have to…

Q: Let G be a graph with n 1 vertices. (a) If x(G) = 1, what can you say about E(G), the edge set of G?…

A: Given : G is a graph with n≥1 vertices We have to answer the following : (a) If χG=1, what can be…

Q: Let H be a graph If H has at least one vertex of degree greater than zero then there are two of even…

A: Given a graph representing H. Statement: If H has at least one vertex of degree greater than zero…

Q: In graphing, prove that at least one of G and it's complement, G', is connected.

A:

Q: Let G be a simple graph on 9 vertices, and assume we know that the sum of all degrees in G is at…

A:

Q: Find a pair, G and H, of isomorphic graphs among the four graphs below, and give the one-to-one…

A: Isomorphism in graphs depends upon the number of edges and vertices present in the graphs. The…

Q: 3. Indicate all the possible clique and maximal clique for the following graph. QA E D

A: A clique of a graph is an included sub graph of a graph that is complete.It is one of the…

Q: Let G = (V, E) be a connected graph. Use induction to show that if G contains exactly |G| = |V|.…

A: Suppose G=(V,E) be a connected graph, also let that G contains one cycle. Then we show that G=V

Q: Let G be a (p, q) graph. Prove that the sum of the degrees of the vertices of G equals twice the…

A:

Q: Let G be a connected graph with n(G) ≥ 2. Show that G at least 2 vertices which are not cut-vertices…

A:

Q: a) Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex…

A: Part(a): G is a simple graph with n vertices and degree of each vertex in G is at least n-12. We…

Q: Let G =(V,E) be a finite graph on n ≥ 3

A:

Q: Let G be an undirected graph with n vertices. Let A(G) be the maximum degree of any vertex in G,…

A:

Q: Prove that for every odd n ≥ 5 there exist a graph of n+1 vertices n of which have degree 3, and the…

A:

Q: In 1 and 2, graphs are represented by drawings. Define each graph formally by specifying its vertex…

A: 1.Note that, a graph consists of two finite sets: a finite, a nonempty set of vertices and a finite…

Q: Let G be a simple graph with n vertices, which is regular of degree d. By considering the number of…

A:

Q: 9. Verify that Euler's formula does not hold for a graph that is not connected: count the vertices,…

A: The given question is related with graph theory. We have to verify that Euler's formula does not…

Q: 2. Let G and H be a graphs. Prove that if G = H, then H= G.

A: Given: It is given that let G and H be a graphs , the objective is to prove that if and then

Q: Give an example of a graph with an infinite number of vertices that has a finite chromatic number.…

A:

Q: 3. The complement of a graph G= (V, E) is the graph G = (V,E) where E = {{r, y} | 1, y E V(G), {r,…

A: a) Let Kn denote the complete graph of n vertices. The complement of the complete graph Kn is…

Q: Graphs G and H are said to be isomorphic if the vertices are in one-to-one correspondence and the…

A: Graph isomorphism: Two graphs G(V,E) and  G'(V',E') are said to be isomorphic if there exists an…

Q: An isomorphism of graphs Gand H is a one to one correspondance of the vertex sets of Gand H. Such…

A: Option 2 is correct.

Q: The graph of g con

A: Consider the graph below

Q: a) Prove that, for all integers K > 5 P(K + 1,5) + P(K,5) = 5P(K,4) b) What is graph, Explain its…

A: We solve only one question at a time. Please submit questions one by one which you want to be…

Q: Let G be a bipartite graph with bipartite blocks with |A|| A = v1, v2, ..., Vm and B = u1, U2, ...,…

A:

Q: attached, thanks

A: To describe the given locus (of z) graphically.

Q: For a given graph G having n vertices and e edges which is connected and has no cycles, which of the…

A: Answer : (A)  Hence,   n = e + 1 is the true statement.

Q: Let G = (V, E) be a k-connected graph, let æ E V, and let B C V\{x} such that |B| > k. Show that…

A: Fan: Let v∈V(G) and let A∈V(G)\{v}. A v,A-fan of size k is a collection of k paths P1,P2,...,Pk so…

Q: If J has at least one vertex of degree greater than zero then there are two of even degree as long…

A: This is a problem of Graph Theory.

Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. a)Prove that…

A:

Q: pigeon hole principle

A: Answer (1) E = { (u,v) , u, v ∈ V | person u and V have met each other}(2) degree of a vertex u of G…

Q: If all the vertices in a graph G have degrees less than or equal to k, then x(G) _ k + 1. Fill in…

A: We can find the upper bound of the chromatic number of the graph if the degree of the vertices is…

Q: Consider an undirected, connected, simple graph with n vertices. What are the minimum and maximum…

A:

Q: 2. Answer the questions with Yes or No, and then explain why yes or why not. Justify your answer. a)…

A: Answer :

Q: 5. For any graph G with six vertices, then G or its complement Ġ contains a triangle.

A: The solution for the above question is as shown below.

Q: prove If G is a plane graph with n vertices, m edges and r regions, then n - m + r = 1 + k(G).

A:

Q: (b) A planar graph has 10 edges and its dual graph has j edges, for some integer j. What is this j?…

A: Use the properties of planar graph and their dual graphs. Use the relation between the number of…

Q: Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to…

A:

Q: 5. Determine whether the given pair of graphs is isomorphic. Us H G

A: We need to check if both have the same vertices and edges, their degree sequence, and the length of…

Q: The number of non-isomorphic simple graphs on 8 vertices, all of which have degree 2, is Select one:…

A:

Q: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

A:

Q: (29) Which of the follwing graphs is Eulerian ? a) c) d) C- e) 圣 at

A:

Q: Prove that in any simple graph G with n vertices and m edges, 2m ≤ n^2 − n.

A: We need to prove that in any simple graph G with n vertices and m edges, 2m≤n2−n

Q: Which of the following graphs contain a subdivision of K3,3 as a subgraph

A: The solution is given below in detail in which we found no graph contains a subdivision of K3, 3 1)

Q: Let G=(V, E) be a graph such that • V| ≥ 3, G has exactly 2 leaves, and in G all the non-leaf…

A:

Q: 9. A bipartite graph G is a simple graph whose vertex set can be partitioned into two disjoint…

A: Since you have asked a question with multiple subparts so as per guidelines we will solve the first…