IN THIS ARTICLE WE AIM TO OBTAIN AN EXPLICIT FORMULA FOR SOME DISTANCE BASED graph INVARIANTS OF DISTANCE-Regular graphS. IN FACT WE OBTAIN FORMULAS FOR WIENER INDEX AND ITS MULTIPLICATIVE VERSION OF A DISTANCE-Regular graph IN TERMS OF ITS INTERSECTION ARRAY AND ITS DISTANCE PARTITION.

THE Regular graph OF IDEALS OF THE COMMUTATIVE RING R, DENOTED BY GREG(R), IS A graph WHOSE VERTEX SET IS THE SET OF ALL NON-TRIVIAL IDEALS OF R AND TWO DISTINCT VERTICES I AND J ARE ADJACENT IF AND ONLY IF EITHER I CONTAINS A J -Regular ELEMENT OR J CONTAINS AN I -Regular ELEMENT. IN THIS TALK, SOME FORMULAS AND BOUNDS FOR THE CLIQUE NUMBER, VERTEX CHROMATIC AND EDGE CHROMATIC NUMBER OF GREG (R) ARE GIVEN. FOR INSTANCE, IT IS SHOWN THAT THE EDGE CHROMATIC NUMBER OF THIS graph EQUALS ITS MAXIMUM DEGREE. SOME APPLICATIONS IN THE RING THEORY ARE ALSO PRESENTED.

IN THIS PAPER, WE STUDY THE PLANARITY, OUTERPLANARITY, AND END-RegularITY OF THE ZERO-DIVISOR graph OF THE RING OF ALL REAL VALUED CONTINUOUS FUNCTIONSC (X), WHICH IS DENOTED BY G (C (X)). ALSO, BY USING THE RING PROPERTIES OF GC (X), THE graph PROPERTIES OF G (C (X)), AND THE TOPOLOGICAL PROPERTIES OF X, WE INVESTIGATE THE END-RegularITY OF THE graph G (C (X)).

IN THIS PAPER, WE CONSIDER A CLASS OF graphS SATISFYING SOME CONDITIONS ON THE NEIGHBOURHOODS OF THEIR VERTICES. IT IS SHOWN THAT ALL Regular SUCH graphS ARE STRONGLY Regular graphS. ALSO, IT IS DETERMINED THAT HOW MANY EDGES CAN THESE graphS HAVE?