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Given any tree T having v vertices of vertex degrees of 1 and 3 only, form an n-expansion by taking n disjoint copies of T and joining corresponding leaves by an n-cycle ...

A method for finding solutions u and v to a linear congruence au+bv=d by constructing a matrix formed by adjoining a vector containing a and b with a unit matrix, M=[a 1 0; b ...

The eigenvalues of a graph are defined as the eigenvalues of its adjacency matrix. The set of eigenvalues of a graph is called a graph spectrum. The largest eigenvalue ...

An (m+1)-dimensional subspace W of an (n+1)-dimensional vector space V can be specified by an (m+1)×(n+1) matrix whose rows are the coordinates of a basis of W. The set of ...

The Fritsch graph is the 9-node planar graph illustrated above that tangles the Kempe chains in Kempe's algorithm and thus provides an example of how Kempe's supposed proof ...

A dissection of a rectangle into smaller rectangles such that the original rectangle is not divided into two subrectangles. Rectangle dissections into 3, 4, or 6 pieces ...

The 10_3 configuration of ten lines intersecting three at a time in 10 points which arises in Desargues' theorem. Its Levi graph is the Desargues graph.

The Diophantine equation x^n+y^n=z^n. The assertion that this equation has no nontrivial solutions for n>2 has a long and fascinating history and is known as Fermat's last ...

A conjecture which relates the minimal elliptic discriminant of an elliptic curve to the j-conductor. If true, it would imply Fermat's last theorem for sufficiently large ...

The orchard-planting problem (also known as the orchard problem or tree-planting problem) asks that n trees be planted so that there will be r(n,k) straight rows with k trees ...