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Adjacency matrix, Order and Valency

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Let A be the adjacency matrix of a regular graph of order v and valency k. Let J be the all-ones matrix of the same order. Show that

A*J = J*A = K*J

"Definition 2.1 A graph r with adjacency matrix A = A(r) is called regular if
there exists a natural number k such that AJ = JA = kJ. The number k is called valency of r."

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Regular Graphs

A graph , is defined by a set of nodes (V) and a set of edges joining those nodes (E).
We write .
Define the order of the graph as the number of nodes in V, let's call this v.
Let's label the nodes as 1, 2...v.
Define the adjacency matrix A with entries such that

That is to say that if there is a link from node i to node j.

Define a simple ...

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