A graph is called k regular if k edges meet at each vertex Let G be a k regular graph (a) Show that the adjacency matrix A of G has k as an eigenvalue (b) Show that if A is primitive, then the other eigenvalues are all less than k in absolute value

Question: A graph is called k-regular if k edges meet at each vertex. Let G be a k-regular graph. (a) Show that the adjacency matrix A

A graph is called k-regular if k edges meet at each vertex. Let G be a k-regular graph.
(a) Show that the adjacency matrix A of G has λ = k as an eigenvalue.

(b) Show that if A is primitive, then the other eigenvalues are all less than k in absolute value.

Step by Step Solution

★★★★★

3.43 Rating (156 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a Each column of A sums to k since each vertex connects to k others Thus A kP ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (2 attachments)

1623_606b0df16fbb7_699282.pdf

180 KBs PDF File

1623_606b0df16fbb7_699282.docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

A connected graph is called a tree if it has no circuits. (a) Find the incidence matrix for each of the following directed trees: (i) (ii) (iii) (iv) (b) Draw all distinct trees with 4 vertices....

(a) Consider the three connected undirected graphs in Fig. 11.11. The graph in part (a) of the figure consists of a cycle (on the vertices u1, u2, u3) and a vertex u4 with edges (spokes) drawn from...

If G = (V, E) is an undirected graph, a subset K of V is called a covering of G if for every edge {a, b} of G either a or b is in K. The set K is a minimal covering if K - {x} fails to cover G for...

Help in Python please ***PLEASE DRAW*** Use the output from running the sample program (GraphClient.py) for graphs to perform the following: Draw the resulting undirected graph and label each vertex....

Your task for this assignment is to identify a spanning tree in a connected undirected weighted graph using C + + . 1 . Implement a spanning tree algorithm using C + + . A spanning tree is an acyclic...

2. Consider the (undirected) graph G having the following vertex set Vand edge set E. V-0,1,2,3,4,5,6,7,8,9 E- 0,1,10,2), 11,2;, 12,4), 12,3), 13,4), (4,5), {5.6,, 14,6, 2,7) a) [2pts] Draw the graph...

6. Graphs. (a) Consider the undirected graph G (V, E), with vertex set V (u, u2, us, us us, uo ur) and the set of edges E-: {(ur, u), (u7, ua), (u7, ua), (u7, t4), (u7,?5), (Ur, te), (ui, t12),...

need typed answer do anyone your wish dont provide code need algorithm 2. A dominating set of a graph is a subset D of vertices such that every vertex that is not in D is adjacent to some vertex in...

JAVA help thank you! The file UDGraph.java contains code for a class UDGraph, an unweighted directed graph represented by a boolean adjacency matrix adjMatrix[][]. For simplicity, vertices are...

you solution this question before but Regarding 5.1 assuming the matrix already input (there is no need to input matrix) Just a simple solution how to write O(n^2) program.... and didn't answer all...

You need some money today and the only friend you have that has any is your miserly' friend. He agrees to loan you the money you need if you make payments of $20 a month for the next six months. In...

rn Morton Company's contribution format income statement for last month is given below: Sales (45,000 units x $21 per unit) Variable expenses Contribution margin Fixed expenses Net operating income...

Which of the following statements is true? Multiple Choice A productievel activity cost will decrease as the number of products produced increases. A productilevel activity cost will increase as the...

In Section 3.6, I said that T scores are designed to have a mean of 50 and a standard deviation of 10 and that the Achenbach Youth Self-Report measure produces T scores. The data in Figure 3.3 do not...

Suppose the expected 2-year risk of a major cardiovascular event in similar but untreated people is 0.028. Test whether the 2-year risk in those treated with a statin is significantly different from...

Controversy persists about whether statins reduce risk of cardiovascular events entirely through their effects on cholesterol levels or whether other pathways, such as reductions in inflammation, are...

What is the estimated incidence rate of testicular cancer among this group of men? Express the rate per 100,000 person-years? Cancer A study was conducted in Sweden to relate the age at surgery for...

At its profit - maximizing output, a pure nondiscriminating monopolist achieves: Group of answer choices both productive efficiency and allocative efficiency. allocative efficiency but not productive...

Use the statistics from the above table to calculate portfolio returns, portfolio variance, and portfolio standard deviation for each investment allocation from Cell A8 to A18

1) 6.18 Is college worth it? Part II: Exercise 6.16 presents the results of a poll where 48% of 331 Americans who decide to not go to college do so because they cannot afford it. (a) (a) Calculate a...