Steinhaus Graphs - A Description


A Steinhaus graph is a special type of undirected graph that is easily represented and computed using an adjacency matrix (1 if adjacent, 0 if not). In the matrix, every place along the diagonal starting at (0,0) down to (n,n) contains a 0. After the first row is given in binary notation, the remaining cells of the matrix may computed using the exlusive or operation. Any one location above the diagonal of zeroes can be computed by looking at the values contained in the cell directly above and the cell northwest of the point being calculated. If one of these two cells but not the other contains a 1, then the cell being computed is filled with a 1. Otherwise, when both positions have the same value, the cell being calculated is filled by a 0. Since Steinhaus graphs are undirected, they are symetric on the diagonal from 0,0 to n,n. As a result of this property, the section of the graph below the diagonal of zeros is filled in when its symetric cell above the diagonal is computed.


The following is an example of a steinhaus graph of six vertices.
011000
100100
100110
011001
001001
000110


History of Steinhaus Graphs

Steinhaus Graphs are named for Hugo Steinhaus who asked if there are Steinhaus triangles containing the same number of 0's ans 1's. A Steinhaus triangle is the upper triangular part of a Steinhaus matrix (excluding the the diagonal). John Molluzz o introduced this class of graphs at the third Western Michigan Conference around 1975. He also examined the complements of Steinhaus graphs. All Steinhaus graphs are connected, except for the ones generated by the all zero string, but the complements c an be disconnected.

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