A directed graph G has 5 vertices, numbered 1 through 5. The 5x5 matrix A is the adjacency matrix for G. The matrices A? and A³ are given below. 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 10 0 0 1 0 0 1 0 0 1 A? = 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 1 0 1

Q: The graph in figure 1 is a sales network, where • s represents a factory producing pop-its, t is the…

A: Given:The graph in figure 1 is a sales network, where: represents a factory producing pop-its, is…

Q: prdinates of point k on the graph could only k

A: We can find and identify the coordinates of K as below

Q: Find all r-regular graphs of order n for a convenient selection of r, n.

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Q: Q4. (a) Prove that for any two simple connected joint graphs G and H, * L(G) UL(H) L(GUH),

A: Let G and H is as below  L(G) =a+cL(H)=b+c

Q: 4. (a) Define f : R×R→R×R by f(x,y) = (2x – y, x – 2y). Prove that f is one-to-one. Describe f-1.…

A: We will use the basic knowledge of functions and set theory to answer both the parts of this…

Q: not use ai please

A: Step 1: It is given that n is even. Let the two components have m and n−m vertices, respectively.…

Q: Prove (Menger) if x, y are vertices of a graph G and xy e E(G), then the minimum size of an x,y-cut…

A: Proof: The proof uses induction on n. The theorem is trivial for n=1. Suppose x and y are separated…

Q: Sketch graph of Quadratic equation along its vertex -x²-2x+3

A: Given: y=-x2-2x+3

Q: TRY IT 1: Given y = x2 -2x - 48, (a) find the vertex, (b) vertex form, (c) axis of symmetry, (d)…

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Q: number of spanning trees of the graph

A: A spanning tree is a subgraph which includes all the vertices of original graph and has number of…

Q: 2n 4. Prove or disprove: for every n-vertex graph G, x(G) ≤ 2w(G) + a (G) (6 + 10.

A: There is n -vertex graph G. For the n vertex graph G, it has to be proven that χ(G) ≤2ω(G) + 2n/α(G)…

Q: whether it is a minimum or a maximu 2.) -4 2 Vertex: (, )__ AY -2 O 1 X

A:

Q: Let n € N with n > 2 and let G₁ = (V, E) be the graph with vertex set V = {₁, V2, ..., V2n} and edge…

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Q: Draw the graph for the funclion Y= SRal and then find (Pp) & (Re)

A: See the attachment

Q: State and prove closed graph theorem

A: The closed graph theorem is given as follows :

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Q: Referring to the graph (G) below, which of the following statement/s is/are TRUE? i. edge BC is…

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Q: 5. Prove or disprove: Let l', and I; be two simple graphs; with V(T)| = m, E(T)=p.V(T)| =n, E(T2)| =…

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Q: Let G(r, s) be the graph with vertex set V= {(a,b) e Zx Z|1<a<r,1<b<s} and edge set = {(a,b), (c,…

A: Given G(r, s) is a graph with vertex set V={(a, b) ∈ℤ×ℤ| 1≤a≤r, 1≤b≤s} and edge set E={{(a, b), (c,…

Q: Find G_1+G_2 and G_10G_2 for the graphs P_2 and C_3 *

A:

Q: Use the formula deg(v) = 2|E(G)| to find the number of edges of the following VEV (G) graphs.…

A: We will be solving part (a) and (b) as mentioned. We know that, for a graph G, the relation between…

Q: The graph of wheel denoted by W, is obtained when an additional vertex is added to cycle Cn, for n >…

A: We need to match each statement (a) - (d) to and a graph (1)-(5) such that the chosen graph…

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A: A cyclic graph has minimum of 3 vertices which is connected in a cycle and a graph is unicyclic…

Q: Find a graph such that w(G) < x(G).

A: Consider the equation

Q: can you please provide explanations

A: 1. Assumption: Start with the assumption that (G1​) and (G2​)are isomorphic graphs. 2. Definition of…

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Q: If G is a graph, then the sum of its vertex degrees is nonnegative and even.

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Q: A graph G is simple if O (c) It is not contain loop or multiple edges. O (d) It is not contains a…

A: