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Bridges between Geometry and Graph Theory
 in Geometry at Work, C.A. Gorini, ed., MAA Notes 53
"... Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another ..."
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Cited by 10 (5 self)
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Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another source of graphs are geometric configurations where the relation of incidence determines the adjacency in the graph. Interesting graphs possess some inner structure which allows them to be described by labeling smaller graphs. The notion of covering graphs is explored.
Isoperimetric Quotient for Fullerenes and Other Polyhedral Cages
"... The notion of Isoperimetric Quotient (IQ) of a polyhedron has been already introduced by Polya. It is a measure that tells us how spherical is a given polyhedron. If we are given a polyhedral graph it can be drawn in a variety of ways in 3D space. As the coordinates of vertices belonging to the same ..."
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The notion of Isoperimetric Quotient (IQ) of a polyhedron has been already introduced by Polya. It is a measure that tells us how spherical is a given polyhedron. If we are given a polyhedral graph it can be drawn in a variety of ways in 3D space. As the coordinates of vertices belonging to the same face may not be coplanar the usual definition of IQ fails. Therefore, a method based on a proper triangulation (obtained from omnicapping) is developed that enables to extend the definition of IQ and compute it for any 3D drawing. The IQ's of fullerenes and other polyhedral cages are computed and compared for their NiceGraph and standard Laplacian 3D drawings. It is shown that the drawings with the maximal IQ values reproduce well the molecular mechanics geometries in the case of fullerenes and exact geometries for Platonic and Archimedean polyhedra.