On the vector space of a-like matrices for tadpole graphs

Consider a simple undirected graph {u100000} with vertex set X. Let MatX(R) denote the R-algebra of matrices with entries in R and with the rows and columns indexed by X. Let A 2 MatX(R) denote an adjacency matrix of {u100000}. For B 2 MatX(R), B is de ned to be A-like whenever the following conditions are satis ed: (i) BA = AB and (ii) for all x y 2 X that are not equal or adjacent, the (x y)-entry of B is zero. Let L denote the subspace of MatX(R) consisting of the A-like elements. The subspace L is decomposed into the direct sum of its symmetric part, and antisymmetric part. This study shows that if {u100000} is T3 n, a tadpole graph with a cycle of order 3 and a path of order n, where n 1, then a basis for L is fI A !g, where A is an adjacency matrix of {u100000}, I is the identity matrix of size jXj, and ! is a block matrix as shown below: In+1 N NT E where N is an (n + 1) 2 zero matrix and E is matrix 0 1 1 0 : If {u100000} is Tm n, where m 4, and n 1, a basis for L is fA Ig