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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 69EQ

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Let G be a connected graph with n ≥ 2 vertices. Let A be the adjacency matrix of G.
Prove that the diameter of G is the least number d such that all the non-diagonal entries
of the matrix A are positive.

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2. Let G be the following graph. V1 V2 V3 V5 VA (a) Let A be the adjacency matrix of G. Find A. (b) Using only A and its powers, determine how many walks of length EXACTLY 3 are there starting at 2 and ending at v4. Explain. (c) Using only A and its powers, determine how many walks of length AT MOST 3 are there starting at 2 and ending at v4. Explain.
Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrixof G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^nis positive.
Let A be an incidence matrix that is associated with a dominance relation. Prove that the matrix A + A2 has a row [column] in which each entry is positive except for the diagonal entry.
Draw the multigraph G whose adjacency matrix is 0 1 2 0 1 1 1 1 A 2 1 0 0 \o 1 0 1/ Show that the maximum number of edges in a graph with n-vertices is *C2
Let M be the incidence matrix and A the adjacency matrix of a graph G.(a) Show that every column sum of M is 2.(b) What are the column sums of A?
Find the adjacency matrices for the directed graphs in (a) and (b).
Use powers of adjacency matrices to determine the number of paths of th e specified length between the given vertices length length 3, v4 to v1
1. Let A be a 6 x 14 matrix (6 rows and 14 columns). Show that the null space of A has dimension at least 8.
Let A be the adjacency matrix of a graph G = (V, E). Moreover, let A' = ,'A' and let æ be the number of zero entries in A'. Prove that G has 1+ connected components. || i=1 (Note: For matrices A, B of size m X n, A + B is the m X n matrix with (A+ B)i,j = Ai,j + Bij) (Hint (part1): start by proving what A' ; represents (to get partial points).) (Hint (part2): assume there are 1 < k < |V] connected components and prove 1 + = k.) ij V
Provide me solution as soon as possible thanks
3. Let G be a siugle graph with vertices M,V2, Vs, V4 aud Vg. Let 0101 10I0I 01011 A - be ts adjacaney matrix. ( Find all paths of leugth 3 be tween Vq and V (b) V and Vu It B. amazon pe here to search
2. Let A Find the projection matrix P, and the projection of b = (1,2,3) onto the column space of A.

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