Must the Graphical Sequence Algorithm always give a connected graph? Assuming the sequence is graphical.
The Graphical Sequence Algorithm is a method used to determine if a given degree sequence is graphical or not, that is, if it can be realized as the degree sequence of a simple graph. The algorithm works by repeatedly checking if the degree sequence can be reduced to a sequence where all the degrees are zero.
Now, to answer your question, no, the Graphical Sequence Algorithm does not always give a connected graph. If the given sequence is graphical, i.e., it can be realized as a graph, the algorithm will output a valid graph. However, this graph may or may not be connected.
The reason for this is that the algorithm only constructs a graph by assigning edges based on the specified degrees for each vertex. The algorithm does not have any information about the connectivity of the graph. It simply determines if a graph with the given degrees is possible or not.
So, while the Graphical Sequence Algorithm guarantees that the degree sequence can be realized as a graph, it does not provide any information about the graph's connectivity. In other words, it is possible to have a graphical sequence that results in a disconnected graph when using the algorithm.