# Algebraic Characterizations of Regularity Properties in Bipartite Graphs

Aida Abiad Monge, C. Dalfó, M.A. Fiol

2 Citations (Scopus)

### Abstract

Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph ?$?$ is distance-regular if and only if its spectral excess (a number that can be computed from the spectrum) equals the average excess (the mean of the numbers of vertices at extremal distance from every vertex). The aim of this paper is to derive new characterizations of regularity and distance-regularity for the more restricted family of bipartite graphs. In this case, some characterizations of (bi)regular bipartite graphs are given in terms of the mean degrees in every partite set and the hoffman polynomial. Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs. Finally, a new version of the spectral excess theorem for bipartite graphs is presented.
Original language English 1223-1231 European Journal of Combinatorics 34 8 https://doi.org/10.1016/j.ejc.2013.05.006 Published - Nov 2013

### Cite this

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title = "Algebraic Characterizations of Regularity Properties in Bipartite Graphs",
abstract = "Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph ?? is distance-regular if and only if its spectral excess (a number that can be computed from the spectrum) equals the average excess (the mean of the numbers of vertices at extremal distance from every vertex). The aim of this paper is to derive new characterizations of regularity and distance-regularity for the more restricted family of bipartite graphs. In this case, some characterizations of (bi)regular bipartite graphs are given in terms of the mean degrees in every partite set and the hoffman polynomial. Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs. Finally, a new version of the spectral excess theorem for bipartite graphs is presented.",
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Algebraic Characterizations of Regularity Properties in Bipartite Graphs. / Abiad Monge, Aida; Dalfó, C.; Fiol, M.A.

In: European Journal of Combinatorics, Vol. 34, No. 8, 11.2013, p. 1223-1231.

TY - JOUR

T1 - Algebraic Characterizations of Regularity Properties in Bipartite Graphs

AU - Dalfó, C.

AU - Fiol, M.A.

N1 - NO DATA USED

PY - 2013/11

Y1 - 2013/11

N2 - Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph ?? is distance-regular if and only if its spectral excess (a number that can be computed from the spectrum) equals the average excess (the mean of the numbers of vertices at extremal distance from every vertex). The aim of this paper is to derive new characterizations of regularity and distance-regularity for the more restricted family of bipartite graphs. In this case, some characterizations of (bi)regular bipartite graphs are given in terms of the mean degrees in every partite set and the hoffman polynomial. Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs. Finally, a new version of the spectral excess theorem for bipartite graphs is presented.

AB - Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph ?? is distance-regular if and only if its spectral excess (a number that can be computed from the spectrum) equals the average excess (the mean of the numbers of vertices at extremal distance from every vertex). The aim of this paper is to derive new characterizations of regularity and distance-regularity for the more restricted family of bipartite graphs. In this case, some characterizations of (bi)regular bipartite graphs are given in terms of the mean degrees in every partite set and the hoffman polynomial. Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs. Finally, a new version of the spectral excess theorem for bipartite graphs is presented.

U2 - 10.1016/j.ejc.2013.05.006

DO - 10.1016/j.ejc.2013.05.006

M3 - Article

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JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

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