Homomorphisms, representations and characteristic polynomials of digraphs

Enumeration of spanning trees of graphs with rotational symmetry

Characteristic polynomials of digraphs having a semi-free action

We present a method to derive the complete spectrum of the lift $$\varGamma ^\alpha $$ of a base digraph $$\varGamma $$, with voltage assignment $$\alpha $$ on a (finite) group G. The method is based on assigning to $$\varGamma $$ a quotient-like matrix whose entries are elements of the group algebra $$\mathbb {C}[G]$$, which fully represents $$\varGamma ^{\alpha }$$. This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalizes some previous results of Lovasz and Babai concerning the spectra of Cayley digraphs.

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The spectra of lifted digraphs

For a graph $$G$$G of order $$n$$n and with eigenvalues $$\lambda _1\geqslant \cdots \geqslant \lambda _n$$?1????n, the HL-index $$R(G)$$R(G) is defined as $$R(G) ={\max }\left\{ |\lambda _{\lfloor (n+1)/2\rfloor }|, |\lambda _{\lceil (n+1)/2\rceil }|\right\} .$$R(G)=max|??(n+1)/2?|,|??(n+1)/2?|. We show that for every connected bipartite graph $$G$$G with maximum degree $$\Delta \geqslant 3$$Δ?3, $$R(G)\leqslant \sqrt{\Delta -2}$$R(G)?Δ-2 unless $$G$$G is the the incidence graph of a projective plane of order $$\Delta -1$$Δ-1. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index.

http://math.ipm.ac.ir/~tayfeh-r/papersandpreprints/mediancovers.pdf

Median eigenvalues of bipartite graphs

In this note, a formula for the characteristic polynomial of any (regular or irregular) graph covering is described.

/pdf/characteristic-polynomials-of-graph-coverings-3iep996wsd.pdf

Characteristic polynomials of graph coverings

In this paper, we give several kinds of characterisations of isomorphic authentication codes by examining a correspondence between optimal authentication codes and some combinatorial designs. The isomorphism classes of some kinds of authentication codes are also enumerated.

/pdf/isomorphism-classes-of-authentication-codes-5fuj6hkq47.pdf

Rongquan Feng

Papers

Characteristic polynomials of graph coverings

Isomorphism classes of authentication codes