Fibonacci-run graphs I: Basic properties

Among the classical models for interconnection networks are hypercubes and Fibonacci cubes. cubes induced subgraphs of obtained by restricting vertex set to those binary strings which do not contain consecutive 1s, counted numbers. Another numbers with a restriction on runlengths. Induced hypercube latter as vertices define Fibonacci-run graphs. They have same number cubes, but fewer edges different graph theoretical properties. We obtain properties graphs including edges, analogue fundamental recursion, average degree vertex, Hamiltonicity, special sequences, they contain. A detailed study sequences is interesting in its own right reported companion paper.