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### Distance-regularised graphs are distance-regular or distance-biregular

Chris D Godsil, John Shawe-Taylor
1987 Journal of combinatorial theory. Series B (Print)
These graphs are shown to either be distance-regular or fall into a family of bipartite graphs called distance-biregular. This family includes the generalised polygons and other interesting graphs.  ...  Despite this increased generahty we are also able to extend much of the basic theory of distance-regular graphs to our wider class of graphs.  ...  In this paper we show that a distance-regularised graph is either distance-regular or is a bipartite semiregular graph from which we can derive two distance-regular graphs.  ...

### Pseudo-distance-regularised graphs are distance-regular or distance-biregular [article]

M. A. Fiol
2012 arXiv   pre-print
In this note, we prove that a pseudo-distance-regular graph around each of its vertices is either 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.  ...  The authors of  used a combinatorial approach to prove that distance-regularised graphs are either distance-regular or distance-biregular. For some properties of the latter, see Delorme  .  ...

### Coverings of complete bipartite graphs and associated structures

John Shawe-Taylor
1994 Discrete Mathematics
Relations with associated distance-biregular graphs are also considered, resulting in the construction of a family of distance-bitransitive graphs.  ...  A construction is given of distance-regular q-fold covering graphs of the complete bipartite graph K qk"pk, where q is the power of a prime number and k is any positive integer.  ...  It has been shown that a distance-regularised graph is either distance-regular or distance-biregular [S].  ...

### The M-matrix group inverse problem for distance-biregular graphs [article]

Aida Abiad, Ángeles Carmona, Andrés M. Encinas, María José Jiménez
2022 arXiv   pre-print
We also show that the two equilibrium arrays characterizing a distance-biregular graph can be expressed in terms of the mentioned equilibrium measures.  ...  In this paper we provide the group inverse of the combinatorial Laplacian matrix of distance-biregular graphs using the so-called equilibrium measures for sets obtained by deleting a vertex.  ...  These graphs are shown to either be distance-regular or distance-biregular. This family includes the generalised polygons and other interesting graphs.  ...

### Automorphism Groups of Primitive Distance-Bitransitive Graphs are Almost Simple

John Shawe-Taylor
1987 European journal of combinatorics (Print)
These graphs are called distance-biregular. It is shown in  that distance-regularised graphs are distance-regular or distance-biregular.  ...  Distance-bitransitive graphs are clearly distance-biregular, as distance-transitive graphs are distance-regular.  ...

### Page 5062 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews
Caccetta (5-CURT) 05 COMBINATORICS 88j:05037 05C75 Godsil, Chris D. (3-SFR); Shawe-Taylor, John (4-LNDHB-S) Distance-regularised graphs are distance-regular or distance-biregular. J. Combin.  ...  Then the distance-regularised graphs introduced in the paper under review are a common generalisa- tion of distance-regular graphs and generalised polygons.  ...

### On \$Q\$-Polynomial Association Schemes of Small Class

Sho Suda
2012 Electronic Journal of Combinatorics
Our method is based on tridiagonal matrices and can be applied to distance-regular graphs as well.  ...  We show an inequality involving the third largest or second smallest dual eigenvalues of \$Q\$-polynomial association schemes of class at least three.  ...  It was proven in  that a distance regularized graph Γ is either distance-regular or distance-biregular.  ...

### On Q-polynomial association schemes of small class [article]

Sho Suda
2012 arXiv   pre-print
Our method is based on tridiagonal matrices and can be applied to distance-regular graphs as well.  ...  We show an inequality involving the third largest or second smallest dual eigenvalues of Q-polynomial association schemes of class at least three.  ...  It was proven in  that a distance regularized graph Γ is either distanceregular or distance-biregular.  ...

### Edge-distance-regular graphs

M. Cámara, C. Dalfó, J. Fàbrega, M.A. Fiol, E. Garriga
2011 Electronic Notes in Discrete Mathematics
Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.  ...  Also, the analogue of the spectral excess theorem for distance-regular graphs is proved, so giving a quasi-spectral characterization of edge-distance-regularity.  ...  They showed that distance-regularised graphs are either distance-regular or distancebiregular. Inspired by this, we introduce the following concept.  ...

### Edge-distance-regular graphs

M. Cámara, C. Dalfó, J. Fàbrega, M.A. Fiol, E. Garriga
2011 Journal of combinatorial theory. Series A
Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.  ...  Also, the analogue of the spectral excess theorem for distance-regular graphs is proved, so giving a quasi-spectral characterization of edge-distance-regularity.  ...  (These graphs are a common generalization of distance-regular graphs and generalized polygons.) Such authors showed that distance-regularised graphs are either distanceregular or distance-biregular.  ...

### Spectral gap in random bipartite biregular graphs and applications

Gerandy Brito, Ioana Dumitriu, Kameron Decker Harris
2021 Combinatorics, probability & computing
We prove an analogue of Alon's spectral gap conjecture for random bipartite, biregular graphs.  ...  A by-product of our main theorem is that random rectangular zero-one matrices with fixed row and column sums are full rank with high probability.  ...  We are grateful to our anonymous reviewers for useful comments and suggestions, and at least in one case for pointing out errors in an earlier version of this manuscript (which have since been fixed).  ...

### Equidistribution and counting under equilibrium states in negative curvature and trees. Applications to non-Archimedean Diophantine approximation [article]

Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin
2019 arXiv   pre-print
Assume that X is a regular or biregular simplicial tree with degrees at least 3, and that Γ is a lattice of X. (1) For every simplicial subtree D of X, we have π˚r σD " ÿ xPV X }µ x } codeg D pxq deg X  ...  For all j, k P N, a graph X is k-regular if the degree of each vertex x P V X is k, and it is pj, kq-biregular if it is bipartite with the elements of the partition of its vertices into two subsets having  ...  Hence, taking s large enough, we may assume that H and γH 1 are disjoint, and that H and H 1 are disjoint or equal.  ...

### Page 1016 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews
Distance-regularised are distance-regular or distance-biregular. J. Combin. Theory Ser. B 43 (1987), no. 1, 14-24.  ...  Cubic distance-regular graphs. J. London Math. Soc. (2) 33 (1986), no. 3, 385-394. (From the text) 884:05135 05C75 Automorphism groups of primitive distance-bitransitive graphs are almost simple.  ...

### Page 368 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews
Ars Combin. 23 (1987), A, 29-34. (R. C. Mullin) 884:05020 05B05 (05C70) (with Shawe-Taylor, John) Distance-regularised graphs are distance-regular or distance-biregular. J. Combin. Theory Ser.  ...  (Notin MR) 93B30 (93B40, 93C10) Godin, Paul On the analytic regularity of weak solutions of analytic systems of conservation laws with analytic data.  ...

### A Dynamic Survey of Graph Labeling

Joseph Gallian
2016 the electronic journal of combinatorics   unpublished
A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the mid 1960s.  ...  has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are  ...  product of C 4 or C 8 and a regular graph is group distance magic; and they show that C 8 × G is also group distance magic for any even-regular graph G.  ...
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