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In this paper, we use the cycles of a cycle basis to replace the faces and obtain an equality of inner faces in Grinberg theorem, called Grinberg equation.

We explain why Grinberg theorem can only be a necessary condition of Hamilton graphs and apply the theorem, to be a necessary and sufficient condition, to simple graphs.

Spacing of primes The Riemann hypothesis holds such a strong allure because it is deeply connected to number theory and, in particular, the prime numbers.

In his 1859 paper, German mathematician Bernhard Riemann investigated the distribution of the prime numbers—or more precisely, the problem "given an integer N, how many prime numbers are there that are smaller than N?

For example, the smallest polyhedral graph that is not Hamiltonian is the Herschel graph on 11 nodes.

All Platonic solids are Hamiltonian (Gardner 1957), as illustrated above.

A Hamiltonian graph on nodes has graph circumference . Testing whether a graph is Hamiltonian is an NP-complete problem (Skiena 1990, p. Rubin (1974) describes an efficient search procedure that can find some or all Hamilton paths and circuits in a graph using deductions that greatly reduce backtracking and guesswork.

This means that there exists a Hamiltonian path if and only if there are edge between consecutive vertices, e.g. If the graph is not connected then there is no Hamiltonian and this algorithm will detect it, because at least one of the consecutive vertices won't be connected (or else the graph will be connected). (Phys.org)—Researchers have discovered that the solutions to a famous mathematical function called the Riemann zeta function correspond to the solutions of another, different kind of function that may make it easier to solve one of the biggest problems in mathematics: the Riemann hypothesis."To our knowledge, this is the first time that an explicit—and perhaps surprisingly relatively simple—operator has been identified whose eigenvalues ['solutions' in matrix terminology] correspond exactly to the nontrivial zeros of the Riemann zeta function," Dorje Brody, a mathematical physicist at Brunel University London and coauthor of the new study, told ABSTRACT: Let G (V, E) be a simple graph with vertex set V and edge set E.A generalized cycle is a subgraph such that any vertex degree is even.

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