1. INTRODUCTION
The phenomenon of representing groups using graphs has been studied theoretically by a number of researchers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The interplay between groups and graphs have been the most famous and productive area of algebraic graph theory. In this paper, we give order divisor graphs of finite groups Un (multiplicative group of integers modulo n) and K4 (Klein’s four group).
2. ORDER DIVISOR GRAPH OF GROUP
2.1. Definition: We call a graph an order divisor graph, denoted by OD(G), if its vertex set is a finite group G and two distinct vertices a and b having different orders are adjacent, provided that o(a) divides o(b) or o(b) divides o(a).
2.2. Order Divisor Graph of Un
The multiplicative
group of integers modulo n is given by 
. The order
divisor graphs of Un for
some n are given as follows:
2.2.1. Order Divisor Graph of U2
The multiplicative group of integers modulo 2 is U2 ={1}. The order divisor graph of U2 shown below is a finite, complete and connected graph. Also the chromatic number is 1.
Fig. 1: OD (U2)
2.2.2. Order Divisor Graph of U3
The multiplicative group of integers modulo 3 is U3 ={1,2}. The order divisor graph of U3 shown below is a finite, complete, connected, bipartite, regular and planar graph. Also the chromatic number is 2.
Fig. 2: OD (U3)
2.2.3. Order Divisor Graph of U5
The multiplicative group of integers modulo 5 is U5= {1,2,3,4}. The order divisor graph of U5 shown below is a finite and connected graph. Also the chromatic number is 3.
Fig. 3: OD (U5)
2.2.4. Order Divisor Graph of U9
The multiplicative group of integers modulo 9 is U9= {1,2,4,5,7,8} . The order divisor graph of U9 shown below is a finite and connected graph. Also the chromatic number is 3.
Fig. 4: OD (U9)
2.2.5. Order Divisor Graph of U13
The multiplicative group of integers modulo 13 is U13= {1,2,3,4,5,6,7,8,9,10,11,12}. The order divisor graph of U13 shown below is a finite and connected graph. Also the chromatic number is 4.
Fig. 5: OD (U13)
2.2.6. Order Divisor Graph of U16
The multiplicative group of integers modulo 16 is U16= {1,3,5,7,9,11,13,15}. The order divisor graph of U16 shown below is a finite and connected graph. Also the chromatic number is 3.
Fig. 6: OD (U16)
2.2.7. Order Divisor Graph of U17
The multiplicative group of integers modulo 17 is U17= {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}. The order divisor graph of U17 shown below is a finite and connected graph.
Fig. 7: OD (U17)
2.3. Order Divisor Graph of K4
The Klein’s four group K4 is given by K4={e,a,b,c}, where a2= b2=c2=e and ab=ba=c, bc=cb= a and ca=ac=b The order divisor graph of K4 shown below is a finite, connected and planar graph. Also the chromatic number is 2.
Fig. 8: OD (K4 )
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Received on 03.10.2017 Modified on 28.12.2017 Accepted on 05.01.2018 ©A&V Publications All right reserved Research J. Science and Tech. 2018; 10(1): 09-12 DOI: 10.5958/2349-2988.2018.00002.5 |
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