Is it possible for a graph to exist that meets these conditions?
For a Graph G, the adjacency matrix has all $1$’s in the first row and all $0$’s in the second row.
What I think: Such a graph cannot exist because if there were $1$’s in the first row, it means that v1 is connected to every single vertex in graph G. However, the second row being all $0$’s indicates that there is no vertex connected to v2, which is a contradiction?
Am I correct?
Also another question, with an adjacency matrix does a loop = 2 in the matrix?
Best Answer
If you are dealing with an undirected graph, the adjcency matrix is a symmetric matrix. Hence, $a_{12} = a_{21}$. Hence, the adjacency matrix cannot represent an undirected graph.
However, if you are talking about adjacency matrix for a directed graph, then the second row can be zero, if there are no edges emanating from the second vertex (depending on the convention you choose).