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.
Answer: (A)
Explanation:
A2 diagonal elements
= (1.1 or 0.0) + (1.1 or 0.0) + ….. + (1.1 or 0.0) [ for n times ]
let some specific node C, if C – A edge present, then 1.1 instead of 0.0 with respect to node A in the above equation
therefore for each edge between C and Vertex, there should be one 1 in the above equation.
Therefore A2 diagonal elements = degree of the vertices of the graph.
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