Combinatorics and Graph Theory

On the Fano Plane Structure, Complete Jordan Basis, and Biorthogonal Projections of Aseven-Vertex Directed Graph

Authors: Peilin Wen
Comments: 8 Pages.

We investigate further algebraic and combinatorial properties of a 7-vertex strongly con- nected directed graph whose adjacency matrix A has characteristic polynomial χA(λ) = λ2(λ − 2)(λ − 1)(λ + 1)3 and Perron root ρ(A) = 2. We establish four main results. First, the14 directed edges of A are shown to correspond to a directed selection from the Fano planeP 2(F2) with one line deleted. Second, we construct a complete Jordan basis of seven integer generalized eigenvectors, yielding the Jordan form J = 1(2)⊕J1(1)⊕J1(−1)⊕J2(−1) ⊕J2(0) with det(V ) = 144. Third, we construct the biorthogonal left-vector basis and seven rank-one projection operators, achieving the Jordan-Chevalley decomposition A = S + N with N 2 = 0. Fourth, we give a spectral computation of the number of closed walks of length L, obtaining NL = 2L +1+3(−1)L, and verify its consistency with the closed-walk equal-weight theorem.
Category: Combinatorics and Graph Theory