Applications of Algebraic concepts of Hydrocarbon in Graph Theory

Abstract

newline xi newlineABSTRACT newline newlineGraph Theory plays a very important role in computer algorithms. Graph theory newlineis one where algebraic methods are applied to the Graphs. Petersen Graph is a special newlinekind of graph and Hydrocarbon is very important area in Organic Chemistry. newline Adjacency matrix of the Petersen Graph whose order is 10 x 10 is derived. newlineCharacteristic polynomial of the Adjacency matrix of the Petersen Graph is and#955; and#8722;and#921; A newline= 0 15 0 75 .......... . 48 10 9 8 7 6 newlineand#955; + and#955; and#8722; and#955; + and#955; + and#955; + + Weighted adjacency newlinematrix of the aliphatic Hydrocarbon and its properties are found. Valancy of the Carbon newlineis 4, and Valancy of the Hydrogen is 1. newline Incidence matrix of the Petersen Graph whose order is 10 x 15 is found . The newlinerank of the incidence matrix of the Petersen Graph is 9. Definition of weighted incidence newlinematrix are given. Incidence matrix of the aliphatic Hydrocarbon and its properties are newlinefound. newline Cycle matrix of the Petersen Graph whose order is 42 x 15 is found . The rank newlineof the Cycle matrix of the Petersen Graph is 6. Cycle matrix of aliphatic newlineHydrocarbon and its properties are found. Cycle matrix of Alkanes does not exist. The newlineCycle matrix of Alkenes is 1. The rank of cycle matrix of Alkynes is 2. newline Edge cutset matrix of the Petersen Graph whose order is 155 x 15 is found. The newlinerank of the Edge cutset matrix of the Petersen Graph is 9. The number of Edge cutsets newlineof Alkanes are respectively 4,7,10 ..3n+1.The number of Edge cutsets of Alkenes newlineare respectively 5,8,11, ..3n + 2. The number of Edge cutsets of Alkynes are newlinerespectively 3,6,9, ..3n. newline Path matrix of Petersen Graph between two vertices whose order is 26 x 15 is newlinefound . Total number of different paths of any two vertices in a Petersen Graph is 26. newlineThe total number of different paths in a Petersen graph is 1,170. newlinexii newline The path matrix of any two vertices of alkanes is row matrix. Atleast one of the newlinepath matrix of any two carbons in alkenes is two rows. Atleast one of the path matrix of newlineany two carbons in alkynes is three rows.