# Research Paper On Graph Theory

When there exists a path that traverses each edge exactly once such that the path begins and ends at the same vertex, the path is known as an Eulerian circuit, and the graph is known as an Eulerian graph.Hamiltonian circuit, along the edges of a dodecahedron (a Platonic solid consisting of 12 pentagonal faces) that begins and ends at the same corner while passing through each corner exactly once.Having considered a surface divided into polygons by an embedded graph, mathematicians began to study ways of constructing surfaces, and later more general spaces, by pasting polygons together.

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Sewer system optimization includes two subproblems: layout optimization and hydraulic design optimization, which can be solved independently or solved simultaneously.

No matter which method is chosen for the solution of the optimization problem, a feasible layout that satisfies the restrictions of the sewer system must be obtained in any step of the solution.

in Indo-Hungarian Pre-Conference School of Conference on Algorithm and Discrete Applied Mathematics (CALDAM 2016) organized by Department of Future Studies, University of Kerala, Thiruvanathpuram, during Feb.18-20, 2016.

in ADMA Pre-Conference Workshop on Recent Advances in Signed Graphs and their Applications, organized by Department of Mathematics, Siddaganga Institute of Technology, Tumkur, Karnataka, during June 06-08, 2016., at BITS Pilani KK Birla Goa Campus, Goa, sponsored by National Board of Higher Mathematics NBHM in collaboration with School of Technology and Computer Science, Tata Institute of Fundamental Research(TIFR) Mumbai, during Jan.

The knight’s tour ( number game: Chessboard problems) is another example of a recreational problem involving a Hamiltonian circuit.

Hamiltonian graphs have been more challenging to characterize than Eulerian graphs, since the necessary and sufficient conditions for the existence of a Hamiltonian circuit in a connected graph are still unknown.

Knowing the number of vertices in a complete graph characterizes its essential nature.

For this reason, complete graphs are commonly designated − 1.

The vertices and edges of a polyhedron form a graph on its surface, and this notion led to consideration of graphs on other surfaces such as a torus (the surface of a solid doughnut) and how they divide the surface into disklike faces.

Euler’s formula was soon generalized to surfaces as Euler characteristic).