A note on nearly Platonic graphs with connectivity one

A note on nearly platonic graphs

We define a nearly platonic graph to be a finite k-regular simple planar graph in which all but a small number of the faces have the same degree. We show that it is impossible for such a graph to have exactly one disparate face, and offer some conjectures, including the conjecture that nearly platonic graphs with two disparate faces come in a small set of families.

A note on vague graphs

In this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. Also, density and balanced irregular vague graphs are discussed and some of their properties are established. Finally we give an application of vague digraphs.

A Note on Atom Bond Connectivity Index

The atom bond connectivity index of a graph is a new topological index was defined by E. Estrada as ABC(G)  uvE (dG(u) dG(v) 2) / dG(u)dG(v) , where G d ( u ) denotes degree of vertex u. In this paper we present some bounds of this new topological index.

Hamiltonian paths on Platonic graphs

We develop a combinatorial method to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle. Platonic graphs with this property are called topologically uniquely Hamiltonian. The same method is used to demonstrate topologically distinct Hamiltonian cycles on the icosahedron graph and to show that a regular graph embeddable on the 2-holed torus is topolog...

A Note on Strongly Quotient Graphs