The sum the adjacency matrix
Task number: 3915
Let \(G\) be a graph and \(A = (a_{ij})_{i,j=1}^n\) its adjency matrix. Determine the sum of elemenets of \(A\), as a function of the number of vertices and edges, i.e. an expression
\(
\sum\limits_{i,j=1}^n a_{ij}.
\)
Hint
What is the sum of one row?Solution
The sum of one row is equal to the degree of the corresponding vertex.
In the grand total, each edge is counted twice.
Answer
The sum of the matrix is \(2|E|\).
