1,678 research outputs found
Pseudo-distance-regularised graphs are distance-regular or distance-biregular
The concept of pseudo-distance-regularity around a vertex of a graph is a
natural generalization, for non-regular graphs, of the standard
distance-regularity around a vertex. In this note, we prove that a
pseudo-distance-regular graph around each of its vertices is either
distance-regular or distance-biregular. By using a combinatorial approach, the
same conclusion was reached by Godsil and Shawe-Taylor for a distance-regular
graph around each of its vertices. Thus, our proof, which is of an algebraic
nature, can also be seen as an alternative demonstration of Godsil and
Shawe-Taylor's theorem
Some results on the structure of multipoles in the study of snarks
Multipoles are the pieces we obtain by cutting some edges of a cubic graph.
As a result of the cut, a multipole has dangling edges with one free end,
which we call semiedges. Then, every 3-edge-coloring of a multipole induces a
coloring or state of its semiedges, which satisfies the Parity Lemma.
Multipoles have been extensively used in the study of snarks, that is, cubic
graphs which are not 3-edge-colorable. Some results on the states and structure
of the so-called color complete and color closed multipoles are presented. In
particular, we give lower and upper linear bounds on the minimum order of a
color complete multipole, and compute its exact number of states. Given two
multipoles and with the same number of semiedges, we say that
is reducible to if the state set of is a non-empty subset of the
state set of and has less vertices than . The function
is defined as the maximum number of vertices of an irreducible multipole with
semiedges. The exact values of are only known for . We prove
that tree and cycle multipoles are irreducible and, as a byproduct, that
has a linear lower bound
Cospectral digraphs from locally line digraphs
A digraph \G=(V,E) is a line digraph when every pair of vertices
have either equal or disjoint in-neighborhoods. When this condition only
applies for vertices in a given subset (with at least two elements), we say
that \G is a locally line digraph. In this paper we give a new method to
obtain a digraph \G' cospectral with a given locally line digraph \G with
diameter , where the diameter of \G' is in the interval .
In particular, when the method is applied to De Bruijn or Kautz digraphs, we
obtain cospectral digraphs with the same algebraic properties that characterize
the formers
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