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Why the adjacency matrix can be used to determine whether or not the graph is connected.

I saw it from the wikipedia.

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If the adjacency matrix can be separated in blocks, then it means there are sets of nodes connected between themselves but not connected to nodes outside their set, so it is not a connected graph.

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The adjacency matrix is just another way of representing the information of a graph. It contains all the relevant information of a graph¹ – each edge corresponds to a 1 somewhere in the adjacency matrix (assuming a binary graph). If you know the adjacency matrix, you know the graph; and vice versa. Therefore all properties of a graph can be deduced from its adjacency matrix, including connectedness.


¹ with the exception of what exactly the nodes are (if you consider this relevant), but that information is obviously irrelevant for connectedness

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