Combined Product of Two RL-graphs and It’s Applications

Abstract

‎This paper introduces the notion of the combined product of two‎ ‎$RL$-graphs‎ ‎while it is an‎ ‎$RL$-graph‎. ‎It is stated that the combined product has commutative properties‎, ‎i.e.‎, ‎$G\boxtimes H$‎ ‎and‎ ‎$H\boxtimes G$ are two isomorphic $RL$-graphs‎. ‎Moreover‎, ‎it is shown under a theorem that two isomorphic‎ ‎$RL$-graphs‎ ‎$G$‎ ‎and‎ ‎$G'$‎ ‎and‎ ‎two isomorphic‎ ‎$RL$-graphs‎ ‎$H$‎ ‎and‎ ‎$H'$‎ ‎have isomorphic combined products‎ ‎$G\boxtimes H$‎ ‎and‎ ‎$G'\boxtimes H'$‎. ‎Further‎, ‎it is investigated the relationships between these graphs and their operations by some notions such as strong‎ ‎$RL$-graph‎, ‎regular $RL$-graph‎, ‎and totally regular $RL$-graph‎. ‎Afterward‎, ‎it is displayed in a theorem that the combined product of two regular ($\alpha$-regular‎, ‎complete‎, ‎connected) $RL$-graphs is a regular ($\alpha$-regular‎, ‎complete‎, ‎connected) $RL$-graph‎. ‎It is also shown in theorems what properties certain types of‎ ‎$RL$-graphs combined will have‎. ‎Also‎, ‎these notions and theorems are clarified by some examples‎. ‎The combined product of two‎ ‎$RL$-graphs has many applications in various fields‎, ‎such as probability sciences‎, ‎urban planning‎, ‎etc‎. ‎In this article‎, ‎only two of these applications‎, ‎which determine the impact of effective factors on people's quality of life‎ ‎and‎ ‎factors effective in raising the production of a factory‎, ‎are stated and they are clarified by an example‎.

Keywords

• $rl$-graph‎
• ‎regular $rl$-graph‎
• ‎combined product of two $rl$-graphs‎