Abstract: The Seven Bridges of Königsberg initiated the idea of circuitry walk. Which gives absolute and finest ideas to how do we reduce or extend the walk from one point to other point. Euler, Hamilton, Cayley, Sylvester and some more mathematicians gave more ideas about graphs. Their works are defining the nature of graphs, walks, measuring the distance between two or more points, solving daily life problems, designing paths and neural circuitries. Graph theory suggesting measurement for geometrical shapes and also non-geometrical shapes. In this paper, I would like to talk about joints (ends) of a certain line by the usage of Pascal’s triangle and the identities of binomial expressions, Invariants of x- joints n dimensional cube, Diagonal relation and difference between vertices, edges, faces, cells and etc...of 2 joints n dimensional cubes. Difference between any two successive odd columns and even columns of x joints n dimensional cube and also I try to generalize Euler’s theorem and graphical equivalent formulae for n-cubes and n-cuboids.