1. Write a Fortran recursive integer function GCD (M,N) which returns the greatest common divisor of positive integer numbers m and n. The recursive definition is as follows:
GCD(m, n) =
GCD(n, m),
if n > m;
m,
if n = 0;
GCD(n, mod(m, n)),
if m > n ∧ n > 0.
where the (Fortran) function mod(m, n) returns the remainder of dividing m by n.
Write also a Fortran program which reads a sequence of integer numbers and uses GCD(M,N) to find tghe greatest common divisor of the numbers in the sequence.
2. Adjacency matrix of a finite directed graph with n vertices is a square matrix A of n × n elements, in which nondiagonal entry ai,j is 1 if there is an edge from node i to node j in the graph, otherwise ai,j is zero. A diagonal entry ai,i is 1 only if there is a loop on node i (i.e. an edge from i to i). Write a Fortran logical function StronglyConn(A, n) which checks if a directed graph described by its adjacency matrix A[1 : n, 1 : n] is strongly connected (a strongly connected graph contains a path connecting any pair of nodes).
Also, write a Fortran program which enters a directed graph, for example as a sequence of edges (i.e., pairs of nodes) and creates the adjacency matrix, invokes StronglyConn and outputs its result. For example, for a 4–node graph with 6 edges:
1,2
1,4
2,3
3,1
3,4
4,2
the adjacency matrix is:
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
This graph is strongly connected.
Hint: To check if there is a path from vertex i to any other vertex of a graph, an auxiliary n-element array X can be used which initially contains a copy of row i from the adjacency matrix A, and to which iteratively are added rows of A indicated by nonzero elements of X. When no new elements are added to X, the iteration ends. All nonzero elements of X indicate those nodes to which there is a path from node i.
To check strong connectivity, the procedure is repeated for all nodes i of the graph.
