The Basis Numbers Of The Line Graphs And The Invertibility Of The LCM Power Matrices

The paper is made up of two parts.In the first part, we discuss the basis number of the line graph of a graph. A basis of the cycle space (G) of a graph G is h-fold if each edge of G occurs in at most h cycles of the basis. The basis number b(G) of G is the least integer h such that (G) has an h-fold basis. In this part we get the basis number of line graph L(G) of some spefial graphs. And we show that let G be a graph, b(L(G)) b(G)+3.In the second part, we discuss the invertibility of LCM power matrices. Let S={x1, x2, ……, xn}be a set of distinct positive integers and m be positive real number. The n X n matrix [S ] = ([xi, Xj] *)is called the LCM power matrix on S, where [xi, Xj] is the least common multiple of Xi and Xj. For m=l, the determinant and invertibility of LCM matrix [S] are discussed extensively. The main result of this part is to show that when m 2 and n 9, the LCM power matrices [S ] on a gcd-closed set is invertible.