Exact two-point resistance, and the simple random walk on the complete graph minus $ N$ edges

An analytical approach is developed to obtain the exact expressions for the two-point resistance, and the total effective resistance of the complete graph minus $N$ edges of the opposite vertices. These expressions are written in terms of certain numbers that we introduced which we call the Bejaia and the Pisa numbers, these numbers are the natural generalizations of the bisected Fibonacci and Lucas numbers. The correspondence between random walks and the resistor networks is then used to obtain the exact expressions for the the first passage and mean first passage times on this graph.