The structure and parameters of a graph assigned to topological spaces

For a set ‎‎X‎ ‎and a ‎topological space ‎‎(X,\tau‎), ‎let‎ ‎‎‎\Gamma_{_X}(\tau)‎ ‎be a‎ ‎graph ‎with ‎vertex ‎set ‎‎‎\tau\setminus \{\emptyset,X\}‎ ‎in ‎which ‎two ‎vertices ‎‎‎A_۱‎ ‎and ‎‎‎A_۲‎ ‎are ‎adjacent ‎just ‎when ‎‎‎A_۱\cup A_۲=X‎‎. In this paper and among some other results, we study the maximum and minimum degrees, the matching number, the chromatic number, the chromatic index, ‎the planarity, the Wiener index and the Zagreb index of ‎‎‎\Gamma_{_X}(\tau)‎ and we determine their exact values in general cases or in some special topological spaces like ‎‎T_۱‎‎‎.

کلیدواژه ها:

Topological space‎ ، ‎graph coloring‎ ، planarity ، Topological indices