Motivated by applications in matrix constructions used the inverse eigenvalue problem for graphs, we study a concept of genericity eigenvectors associated with given spectrum and connected graph. This generalizes established notion nowhere-zero eigenbasis. Given any simple graph G on n vertices no multiple eigenvalues, show that family eigenbases symmetric matrices this is generic, strengthenin...

We prove that every line graph of a 4-edge-connected graph is Z3-connected. In particular, every line graph of a 4-edge-connected graph has a nowhere zero 3-flow.

We study reducibility for nowhere-zero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counter-example to some conjecture. We derive algebraic proofs of reducibility. We define variables which in some sense count the number of nowhere-zero flows of certain type in a graph and then deduce equalities and inequalities that must hold ...

An orientation of a graph is acyclic (totally cyclic) if and only if it is a “positive orientation” of a nowhere-zero integral tension (!ow). We unify the notions of tension and !ow and introduce the so-called tension-!ows so that every orientation of a graph is a positive orientation of a nowhere-zero integral tension-!ow. Furthermore, we introduce an (integral) tension-!ow polynomial, which g...

The concepts of (k, d)-coloring and the star chromatic number, studied by Vince, by Bondy and Hell, and by Zhu are shown to reflect the cographic instance of a wider concept, that of fractional nowhere-zero flows in regular matroids. c © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 155–161, 1998

We construct an “orbital Tutte polynomial” associated with a dual pair M and M∗ of matrices over a principal ideal domain R and a group G of automorphisms of the row spaces of the matrices. The polynomial has two sequences of variables, each sequence indexed by associate classes of elements of R. In the case where M is the signed vertex-edge incidence matrix of a graph Γ over the ring of intege...

We study quasipolynomials enumerating proper colorings, nowherezero tensions, and nowhere-zero flows in an arbitrary CW-complex X, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may be either modular (with values in Z/kZ for some k) or integral (with values in {−k + 1, . . . , k − 1}). We obtain deletion-contraction recurrences and closed ...