The sphericity of a graph is the smallest dimension such that there exists a mapping such that if and only if . The following theorem is due to Frankl and Maehara and is a nice application of random projections. Here, . Theorem [Frankl-Maehara 1988]: Let be a graph with minimum adjacency eigenvalue where and suppose . […]
This formula is well known and is given as an exercise for integration by parts. Here is a nice alternative proof I came across due to Inna Zakharevich. . Now differentiate times wrt to and set .
Translation Honor to those who in their lives have defined and guard their Thermopylae. Never stirring from duty; just and upright in all their deeds, yet with pity and compassion too; generous when they are rich, and when they are poor, again a little generous, again helping as much as they can; always speaking the […]
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