Walks on Directed Graphs and Matrix Polynomials

We give a matrix generalization of the family of exponential polynomials in one variable φk(x). Our generalization consists of a matrix of polynomials Φk(X)=(Φ(k)i, j(X))ni, j=1 depending on a matrix of variables X=(xi, j)ni, j=1. We prove some identities of the matrix exponential polynomials which generalize classical identities of the ordinary exponential polynomials. We also introduce matrix generalizations of the decreasing factorial (x)k=x(x−1)(x−2)…(x−k+1), the increasing factorial (x)(k)=x(x+1)(x+2)…(x+k−1), and the Laguerre polynomials. These polynomials have interesting combinatorial interpretations in terms of different kinds of walks on directed graphs.