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Let z_0 be a point in a simply connected region R!=C, where C is the complex plane. Then there is a unique analytic function w=f(z) mapping R one-to-one onto the disk |w|<1 ...

A subset X of R^n is star convex if there exists an x_0 in X such that the line segment from x_0 to any point in X is contained in X. A star-shaped figure is star convex but ...

If W is a simply connected, compact manifold with a boundary that has two components, M_1 and M_2, such that inclusion of each is a homotopy equivalence, then W is ...

The above topological structure, composed of a countable union of compact sets, is called Alexander's horned sphere. It is homeomorphic with the ball B^3, and its boundary is ...

If f is continuous on a closed interval [a,b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that ...

The maximum leaf number l(G) of a graph G is the largest number of tree leaves in any of its spanning trees. (The corresponding smallest number of leaves is known as the ...

Define a = d(u,v)d(w,x) (1) b = d(u,w)d(v,x) (2) c = d(u,x)d(v,w), (3) where u, v, w, and x are vertices of a graph and d(i,j) is the graph distance between vertices i and j. ...

Let a random n×n (0,1)-matrix have entries which are 1 (with probability p) or 0 (with probability q=1-p). An s-cluster is an isolated group of s adjacent (i.e., horizontally ...

The algebraic connectivity of a graph is the numerically second smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of a graph G. In other ...

A bridged graph is a graph that contains one or more graph bridges. Examples of bridged graphs include path graphs, ladder rung graphs, the bull graph, star graphs, and ...