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### Distance-regular digraphs of girth 4

H Enomoto, R.A Mena
1987 Journal of combinatorial theory. Series B (Print)
Thus, g = 4 (and d = 3) is the smallest case. II. PARAMETRIC CONSIDERATIONS Let g be a distance-regular digraph with g = 4 and d = 3.  ...  Then the subdigraph X is (p -1 )-regular on 2~ -1 vertices, and the girth of X is at least 4. This means that X is a (regular) tournament. However, a strongly connected tournament has girth 3.  ...

### Distance-transitive and distance-regular digraphs

R.M. Damerell
1981 Journal of combinatorial theory. Series B (Print)
We adopt the definition of a distance-transitive digraph given by Lam.  ...  Finally we mention an alternative definition of distance transitivity, which in our opinion merits study.  ...  In Section 3 we shall compare the theory of distance-regular digraphs with the theory  for undirected graphs. In Section 4 we prove the second main theorem.  ...

### On nonsymmetric P- and Q-polynomial association schemes

Akihiro Munemasa
1991 Journal of combinatorial theory. Series B (Print)
A distance-regular digraph with diameter 2 and girth 3 can be constructed from a skew Hadamard matrix; distance-regular digraphs with diameter 3 and girth 4 have been constructed by Liebler and Mena [6  ...  These distance-regular digraphs are short digraphs; i.e., the diameter is one less than the girth.  ...  A distance-regular digraph with diameter 2 and girth 3 can be constructed from a skew Hadamard matrix; distance-regular digraphs with diameter 3 and girth 4 have been constructed by Liebler and Mena [  ...

### A spectral excess theorem for normal digraphs

G. R. Omidi
2015 Journal of Algebraic Combinatorics
Hence such a digraph is strongly regular with girth g = 2 or g = 3. MSC(2010): Primary: 05C50; Secondary: 05E30.  ...  The spectral excess theorem, due to Fiol and Garriga in 1997, is an important result, because it gives a good characterization of distance-regularity in graphs.  ...  Omidi for his helpful suggestions during the preparation of the paper.  ...

### Commutative weakly distance-regular digraphs of girth 2

Kaishun Wang
2004 European journal of combinatorics (Print)
We introduce some constructions of weakly distance-regular digraphs of girth 2, and prove that a certain quotient digraph of a commutative weakly distance-transitive digraph of girth 2 is a distancetransitive  ...  Finally, all commutative weakly distance-regular digraphs of valency 3 and girth 2 are classified.  ...  This research was supported by National Natural Science Foundation of China (10301005, 10171006).  ...

### A family of weakly distance-regular digraphs of girth 2

Kaishun Wang
2006 Discrete Mathematics
A family of commutative weakly distance-regular digraphs of girth 2 was classified in [K. Wang, Commutative weakly distanceregular digraphs of girth 2, European J. Combin. 25 (2004) 363-375].  ...  In this paper, we classify this family of digraphs without the assumption of commutativity.  ...  Acknowledgements The author thanks the referees for their close reading of the manuscript and for valuable suggestions that improved the presentation of the paper.  ...

### On the Spectrum of a Weakly Distance-Regular Digraph

F. Comellas, M.A. Fiol, J. Gimbert, M. Mitjana
2001 Electronic Notes in Discrete Mathematics
In particular, a (strongly) distance-regular digraph Γ is stable, which means that A i = A g−i , for each i = 1, . . ., g − 1, where g denotes the girth of Γ.  ...  We consider the class of digraphs characterized by such a weaker condition, referred to as weakly distance-regular digraphs, and show that their spectrum can also be obtained from a smaller 'quotient digraph  ...  Introduction The concept of a distance-regular digraph was introduced by Damerell  as a generalization of the notion of distance-transitivity given by Lam  .  ...

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

G.R. Omidi
2013 arXiv   pre-print
It is known that every distance-regular digraph is connected and normal. An interesting question is: when is a given connected normal digraph distance-regular?  ...  Motivated by this question first we give some characterizations of weakly distance-regular digraphs.  ...  Using Equation 4 Theorem 4. 4 . 4 A connected normal digraph Γ is a geodetic distance-regular digraph if and only if Q d 2 = n.  ...

### Weakly distance-regular digraphs

Kaishun Wang, Hiroshi Suzuki
2003 Discrete Mathematics
We consider the following generalization of distance-regular digraphs.  ...  We give some constructions of weakly distance-regular digraphs and discuss the connections to association schemes. Finally, we determine all commutative weakly distance-regular digraphs of valency 2.  ...  Damerell  generalized this concept to that of distance-regular digraphs.  ...

### Minimum edge cuts of distance-regular and strongly regular digraphs [article]

S. Ashkboos, G.R. Omidi, F. Shafiei, K. Tajbakhsh
2017 arXiv   pre-print
In this paper, we show that the edge connectivity of a distance-regular digraph Γ with valency k is k and for k>2, any minimum edge cut of Γ is the set of all edges going into (or coming out of) a single  ...  Moreover we show that the same result holds for strongly regular digraphs. These results extend the same known results for undirected case with quite different proofs.  ...  Moreover distance-regular digraphs with girth g = 2 are precisely the distance-regular graphs.  ...

### Page 6644 of Mathematical Reviews Vol. , Issue 92m [page]

1992 Mathematical Reviews
Distance-regular digraphs come in two varieties: short digraphs, where the diameter is one less than the girth; and long ones, where the diameter equals the girth. R. M.  ...  In this paper, the author shows that the intersection numbers of short distance-regular digraphs of girth 6 can be written in terms of three parameters.  ...

### Further Topics in Connectivity [chapter]

Camino Balbuena, Josep Fàbrega, Miguel Fiol
2013 Discrete Mathematics and Its Applications
EXAMPLE E2: Figure 4 .7.2 shows a 2-regular maximally connected digraph G that is not super-.  ...  F71: Let G be a d-regular graph with n vertices, D 2 > 1, and Laplacian eigenvalues ✓ Figure 4 4 Figure 4.7.6: F is a fragment.  ...  Distance Connectivity Here we consider a generalization of the concepts of connectivity and edge-connectivity of a (di)graph, introduced in [FiFa94] and [BaCaFi96] , which takes into account the distance  ...

### On the distance connectivity of graphs and digraphs

M.A. Fiol, J. Fàbrega
1994 Discrete Mathematics
This paper studies some results on the distance connectivities of digraphs and bipartite digraphs.  ...  Let G=( V, E) be a digraph with diameter D # 1. For a given integer 1 t. The t-distance edge-connectivity of G is defined analogously.  ...  (a) Every bipartite digraph with distance connectivity lc(3)>6 has maximum connectivity. Corollary 3. 4 . 4 Let G be a bipartite graph with minimum degree 6 > 1, (even) girth g and diameter D.  ...

### Short Cycles in Digraphs with Local Average Outdegree at Least Two

Jian Shen
2003 Electronic Journal of Combinatorics
Suppose \$G\$ is a strongly connected digraph with order \$n\$ girth \$g\$ and diameter \$d\$.  ...  In particular, we prove that, for any digraph \$G\$ of order \$n\$, if either(1) \$G\$ has minimum outdegree \$1\$ and \$\deg^+(u) +\deg^+(v) \ge 4\$ for all arcs \$(u,v)\$, or(2) \$\deg^+(u) +\deg^+(v) \ge 3\$ for  ...  The diameter of G, denoted d(G), is the maximum distance among all ordered pairs of vertices in G. In 1970, Behzad  proved that the girth of any 2-regular digraph of order n is at most n/2 .  ...

### From Subkautz Digraphs to Cyclic Kautz Digraphs

C. DALFÓ
2018 Journal of Interconnection Networks (JOIN)
Finally, we provide a lower bound on the girth of CK(d, ) and sK(d, ). Mathematics Subject Classifications: 05C20, 05C50.  ...  Closely related with these, the cyclic Kautz digraphs CK(d, ) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed.  ...  Observe that, since CK(d, 3) is the line digraph of sK(d, 2), the respective mean distance satisfies the inequality δ < δ * , in concordance with the results by Fiol, Yebra, and Alegre  .  ...
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