Respectable Graphs (Report‪)‬

[section] 1. Introduction Let A(X) (or simply A, if X is clear from the context) be the adjacency matrix of a graph X. The set of all polynomials in A with coefficients from the field of complex numbers C forms an algebra called the adjacency algebra of X, denoted by A(X). Let dim(A(X)) denote the dimension of A(X) as a vector space over C. It is easy to see that dim(A(X)) is equal to degree of the minimal polynomial of A. Since dim(A(X)) is also equal [absolute value of spec(A)] where spec(A) denote the set of all distinct eigenvalues of A and [absolute value of B] denote the cardinality of the set B.