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Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?
- a)The diagonal entries of A2 are the degrees of the vertices of the graph.
- b)If the graph is connected, then none of the entries of An-1 + In can be zero.
- c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.
- d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.
Correct answer is option 'A'. Can you explain this answer?
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Which of the properties hold for the adjacency matrix A of a simple un...
A) The diagonal entries of A^2 are not necessarily the degrees of the vertices of the graph. The diagonal entries of A^2 represent the number of length-2 paths between each pair of vertices in the graph. The degrees of the vertices are represented by the diagonal entries of A.
b) If the graph is connected, then all entries of A^(n-1) + A^(n-2) + ... + A + I (where I is the identity matrix) are positive. This is known as the matrix-tree theorem, and it states that the sum of these matrices (up to A^(n-1)) gives the Laplacian matrix of the graph, and the determinant of the Laplacian matrix is equal to the number of spanning trees in the graph. Therefore, if the graph is connected, the determinant of the Laplacian matrix is non-zero, which means none of the entries of A^(n-1) + A^(n-2) + ... + A + I can be zero.
c) If the sum of all the elements of A is at most 2(n - 1), it does not necessarily mean that the graph is acyclic. The sum of all the elements of A represents the total number of edges in the graph. If this sum is at most 2(n - 1), it means that the graph has at most 2(n-1) edges, which is the maximum number of edges in a tree. However, the graph could still have cycles if it is not a tree.
d) If there is at least a 1 in each of A, then it means that each vertex is connected to at least one other vertex in the graph. This does not necessarily imply any specific property of the graph, as it could still have cycles and be a connected graph.
b) If the graph is connected, then all entries of A^(n-1) + A^(n-2) + ... + A + I (where I is the identity matrix) are positive. This is known as the matrix-tree theorem, and it states that the sum of these matrices (up to A^(n-1)) gives the Laplacian matrix of the graph, and the determinant of the Laplacian matrix is equal to the number of spanning trees in the graph. Therefore, if the graph is connected, the determinant of the Laplacian matrix is non-zero, which means none of the entries of A^(n-1) + A^(n-2) + ... + A + I can be zero.
c) If the sum of all the elements of A is at most 2(n - 1), it does not necessarily mean that the graph is acyclic. The sum of all the elements of A represents the total number of edges in the graph. If this sum is at most 2(n - 1), it means that the graph has at most 2(n-1) edges, which is the maximum number of edges in a tree. However, the graph could still have cycles if it is not a tree.
d) If there is at least a 1 in each of A, then it means that each vertex is connected to at least one other vertex in the graph. This does not necessarily imply any specific property of the graph, as it could still have cycles and be a connected graph.
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Which of the properties hold for the adjacency matrix A of a simple un...
Option 1: The diagonal entries of A2 are the degrees of the vertices of the graph.
True, Assume the adjacency matrix representation of the undirected graph is "A". A2= A x A
Diagonal elements of A2 are the degree of vertices or (Aij2) represents the degree of node i.
True, Assume the adjacency matrix representation of the undirected graph is "A". A2= A x A
Diagonal elements of A2 are the degree of vertices or (Aij2) represents the degree of node i.
K is adjacency matrix is,
Degree sequence of (p,q,r)= (1,2,1)
Diagonal elements= (1,2,1)
Diagonal elements= (1,2,1)
Option 2: If the graph is connected, then none of the entries of An-1 + In can be zero.
False,
Take the following connected graph with n=3,
False,
Take the following connected graph with n=3,
K is adjacency matrix is,
We can see that Kn-1+In have some entries as zero.
Option 3: If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.
False, Consider following acyclic graph with n=5
Option 3: If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.
False, Consider following acyclic graph with n=5
A is adjacency matrix is,
So sum of all elements in A=8
here n=5, So 2(n-1)=2(5-1)=8
Here above graph satisfies the condition of the graph but the above graph is not acyclic.
Option 4: If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.
False, Consider following acyclic graph with n=5
here n=5, So 2(n-1)=2(5-1)=8
Here above graph satisfies the condition of the graph but the above graph is not acyclic.
Option 4: If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.
False, Consider following acyclic graph with n=5
Consider the above graph in A all rows and columns have at least A 1 but it disconnected the graph. Hence the given option is false.
Hence the correct answer is the diagonal entries of A2 are the degrees of the vertices of the graph.
Hence the correct answer is the diagonal entries of A2 are the degrees of the vertices of the graph.
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Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer?
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Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer?.
Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer?.
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Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer?, a detailed solution for Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?a)The diagonal entries of A2 are the degrees of the vertices of the graph.b)If the graph is connected, then none of the entries of An-1 + In can be zero.c)If the sum of all the elements of A is at most 2(n - 1), then the graph must be acyclic.d)If there is at least a 1 in each of A’s rows and columns, then the graph must be connected.Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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