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### Regular graphs with excess one

Eiichi Bannai, Tatsuro Ito
1981 Discrete Mathematics
It is she\-,ln that there exists no regular graph with excess __ = 1 and girth 2r + la 5. Let r be a regular graph with valency k and girth 2r + 1 (odd). Let H be the number of the vertices.  ...  There exists no regular graph with excess e = 1 and girth 2r + 12 5 I We remark that if e = 1 and r = 1, then r is a cc&tail. party graph (namely, r= K2.2,. . , -).  ...  There exists no regular graph with valency k and diameter r (2 2) and with defect I (except for an ordinary 4-gon with k = 2 and r = 2).  ...

### An odd characterization of the generalized odd graphs

Edwin R. van Dam, Willem H. Haemers
2011 Journal of combinatorial theory. Series B (Print)
We show that any connected regular graph with d+1 distinct eigenvalues and odd-girth 2d+1 is distance-regular, and in particular that it is a generalized odd graph.  ...  Spectral excess theorem. Let Γ be a connected regular graph with d + 1 distinct eigenvalues. Then Γ is distance-regular if and only if the average excess equals the spectral excess.  ...  In general it is not true that any connected regular graph with diameter D and odd-girth 2D + 1 is a generalized odd graph.  ...

### An Odd Characterization of the Generalized Odd Graphs

Edwin van Dam, Willem H. Haemers
2010 Social Science Research Network
We show that any connected regular graph with d + 1 distinct eigenvalues and odd-girth 2d + 1 is distance-regular, and in particular that it is a generalized odd graph.  ...  Spectral Excess Theorem. Let Γ be a connected regular graph with d+1 distinct eigenvalues. Then Γ is distance-regular if and only if the average excess equals the spectral excess.  ...  To show the claimed characterization, we shall use the so-called spectral excess theorem due to Fiol and Garriga [10] . Let Γ be a connected k-regular graph with d + 1 distinct eigenvalues.  ...

### A spectral excess theorem for nonregular graphs

Guang-Siang Lee, Chih-wen Weng
2012 Journal of combinatorial theory. Series A
The spectral excess theorem asserts that the average excess is, at most, the spectral excess in a regular graph, and equality holds if and only if the graph is distance-regular.  ...  For application, we demonstrate that a graph with odd-girth 2d + 1 must be distance-regular, generalizing a recent result of van Dam and Haemers.  ...  Introduction Throughout this paper, let G = (V G, EG) be a connected graph on n vertices, with diameter D, adjacency matrix A, and distance function ∂.  ...

### The Laplacian spectral excess theorem for distance-regular graphs

E.R. van Dam, M.A. Fiol
2014 Linear Algebra and its Applications
The spectral excess theorem states that, in a regular graph G, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess  ...  In this note we prove the corresponding result by using the Laplacian spectrum without requiring regularity of G.  ...  The authors thank a referee for comments on an earlier version.  ...

### A short proof of the odd-girth theorem [article]

Edwin R. van Dam, Miquel Angel Fiol
2012 arXiv   pre-print
Recently, it has been shown that a connected graph Γ with d+1 distinct eigenvalues and odd-girth 2d+1 is distance-regular. The proof of this result was based on the spectral excess theorem.  ...  In this note we present an alternative and more direct proof which does not rely on the spectral excess theorem, but on a known characterization of distance-regular graphs in terms of the predistance polynomial  ...  Introduction The spectral excess theorem [10] states that a regular graph Γ is distance-regular if and only if its spectral excess (a number which can be computed from the spectrum of Γ) equals its average  ...

### 1-factor covers of regular graphs

sarah-marie belcastro, Michael Young
2011 Discrete Applied Mathematics
In particular, we classify cubic graphs such that every minimal 1-factor cover is also a 1-factorization, and also classify simple regular bipartite graphs with this property.  ...  We consider minimal 1-factor covers of regular multigraphs, focusing on those that are 1-factorizations.  ...  The results of this paper on regular graphs G with exc max (G) = 0 lead to more general questions about the excess range of regular graphs. The Petersen Graph has excess range [2, 2] .  ...

### The Spectral Excess Theorem for Distance-Biregular Graphs

Miquel Àngel Fiol
2013 Electronic Journal of Combinatorics
The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess.  ...  A bipartite graph \$\Gamma\$ is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to.  ...  The author would like to thank the anonymous reviewer for helpful comments and corrections on a first version of this paper.  ...

### Graphs of arbitrary excessive class

G. Mazzuoccolo, M. Young
2011 Discrete Mathematics
We show that there exists a family of r-regular graphs of arbitrarily large excessive index for each integer r greater than 3.  ...  Furthermore, we answer a question in Bonisoli and Cariolaro (2007) [1] showing that all the positive integers can be attained as excessive classes of regular graphs.  ...  r-regular graphs of arbitrary large excessive class In this section we construct a family of r-regular graphs with excessive index arbitrarily large. We treat the cases of even and odd r separately.  ...

### The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues [article]

M.A. Fiol
2014 arXiv   pre-print
In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue  ...  This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser  ...  Let Γ be a regular graph with n vertices, spectrum sp Γ as above, and mean excess k d .  ...

### Variations of the spectral excess theorem for normal digraphs [article]

G.R. Omidi
2013 arXiv   pre-print
Distance regularity of a digraph (also a graph) is in general not determined by its spectrum.  ...  Specially we show that whether a given connected digraph to be weakly distance-regular only depends on the equality for two invariants.  ...  On the other hand, among the triangle-free connected regular graphs with diameter two there are many graphs that are not strongly regular.  ...

### A simple proof of the spectral excess theorem for distance-regular graphs

M.A. Fiol, S. Gago, E. Garriga
2010 Linear Algebra and its Applications
The spectral excess theorem provides a quasi-spectral characterization for a (regular) graph Γ with d + 1 distinct eigenvalues to be distance-regular graph, in terms of the excess (number of vertices at  ...  In this paper we present a new simple projection method based in a global point of view, and where the mean excess plays an essential role.  ...  Acknowledgements The authors wish to thank the support by the Ministry of Science and Technology (Spain) with the European Regional Development Fund under Projects MTM2005-08990-C02-01 and TEC2005-03575  ...

### Preface: Geometric and algebraic combinatorics

Edwin R. van Dam, Willem H. Haemers
2012 Designs, Codes and Cryptography
On the other hand, many association schemes are interesting objects in themselves. This includes the strongly regular and distance-regular graphs.  ...  On one hand it can be a tool for a better understanding of combinatorial objects, such as error correcting codes, block designs, point-line incidence geometries, and permutation groups.  ...  They show that a distance-regular graph with large c 2 is bipartite, and that if the excess is small then the distance-regular graph is antipodal. 6.  ...

### The spectral excess theorem for distance-biregular graphs [article]

M.A. Fiol
2013 arXiv   pre-print
The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess.  ...  A bipartite graph is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to.  ...  Introduction The spectral excess theorem, due to Fiol and Garriga [12] , states that a regular (connected) graph Γ is distance-regular if and only if its spectral-excess (a number which can be computed  ...

### On some approaches to the spectral excess theorem for nonregular graphs [article]

M.A. Fiol
2012 arXiv   pre-print
The Spectral Excess Theorem (SPET) for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess.  ...  corresponding distance-regularity property.  ...  One of such versions concerns with the so-called pseudo-distance-regularity [17] , which is a natural generalization, for nonregular graphs, of the standard distance-regularity [1, 2] .  ...
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