Algebraic properties of a digraph and its line digraph

Abstract

Let G be a digraph, LG its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. In addition, we study the spectra of those matrices and obtain a relationship between their characteristic polynomials that allows us to relate properties of G and LG, specifically the number of cycles of a given length. Ker(A -XI) c Ker(A -XI) 2 C ... Ker (A -AI) r > = Ker(A -XI) r * +1 = ... J. Inter. Net. 2003.04:377-393. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/22/15. For personal use only.

On bipartite‐mixed graphs

“…(This is because G is the line digraph of C 4 , see Balbuena et al. .) In fact, the mixed graph of Figure B is cospectral with G , and it can be obtained by applying a recent method to obtain cospectral digraphs (see Dalfó and Fiol ).…”

“…The spectrum of this mixed graph is the same as the cycle 4 plus four more 0's, that is, sp = {2, 0 6 , −2}. (This is because is the line digraph of 4 , see Balbuena et al [1].) In fact, the mixed graph of Figure 2B is cospectral with , and it can be obtained by applying a recent method to obtain cospectral digraphs (see Dalfó and Fiol [5]).…”

On d-Fibonacci digraphs

“…Since N (d, i) has no meaning for i ≤ 0, an alternative way of stating Proposition 2.1, would be to distinguish two cases: For k ≥ d, N (d, k + 1) is computed by using 4; and, for 1 ≤ k < d, we have Notice that, from ( 7), we proved that not only the total number of vertices of F (d, k) but also those vertices whose sequences end with a given digit j ∈ [0, d − 1] satisfy similar recurrence relations as those of d-step Fibonacci numbers in (2). More precisely, Notice that j = 0 behaves as j = d(≡ 0 mod d).…”

“…For instance, the 2-Fibonacci digraphs F (2, k) with k ≤ 4 and 2, 3, 5, 8 vertices, are shown in Figure 1, whereas the 1-Fibonacci digraphs F (d, 1) on d vertices, with d ∈ [2,5], are depicted in Figure 3.…”

Iterated line digraphs are asymptotically dense

“…Moreover, if G is a digraph (different from a directed cycle) with diameter k and maximum eigenvalue λ 0 , then its -iterated line digraph L (G) has diameter k = k + (see Fiol, Yebra, and Alegre [3,4]), maximum eigenvalue λ 0 (the line digraph technique preserves all the eigenvalues, see Balbuena, Ferrero, Marcote, and Pelayo [1]), and number of vertices…”