day 106: the topic of the night is graphs

Salut, students.

The content below is courtesy of the venerable It’s a little peek into the geometric realm of mathers, graphers, and pathers. (Not panthers.)

A regular graph of degree r is a graph all of whose vertices have the same number of edges incident on them. Regular graphs therefore have a high degree of symmetry, and include a number of particularly beautiful examples, such as those illustrated above.

There is a class of graphs known as strongly regular graphs that are even more symmetric than regular graphs. In a strongly regular graph, not only does each vertex have the same number of neighbors (usually denoted k), but every adjacent pair of vertices has the same number (usually denoted lambda) of common neighbors, and every nonadjacent pair of vertices has the same number (usually denoted mu) of common neighbors. A strongly regular graph on nu vertices is therefore said to have parameters (nu, k, lambda, mu). Strongly regular graphs are very special mathematical objects that have connections to other areas of mathematics, including group theory and statistics. One class of strongly regular graphs is the so-called lattice graphs Ln, whose vertices can be thought of as squares on an n×n chessboard, and whose edges can be thought of as pairs of squares that are connected by a possible move by a rook chess piece. It turns out that the lattice graphs are strongly regular, in particular L4, which is strongly regular with parameters (16, 6, 2, 2). Shrikhande established that there exists exactly one other strongly regular graph with those parameters, and that graph, illustrated below in another embedding, now bears his name.


Good night math lovers, graph lovers, and path lovers. (And laugh lovers.) Pursue symmetry relentlessly, in life and in dreams. Keep on keeping on, with a fire and fight.

KT out


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