Shafiei, Fateme A spectral excess theorem for digraphs with normal Laplacian matrices. (English) Zbl 1463.05354 Trans. Comb. 7, No. 3, 19-28 (2018). Summary: The spectral excess theorem, due to M. A. Fiol and E. Garriga [J. Comb. Theory, Ser. B 71, No. 2, 162–183 (1997; Zbl 0888.05056)], is an important result, because it gives a good characterization of distance-regularity in graphs. Up to now, some authors have given some variations of this theorem. Motivated by this, we give the corresponding result by using the Laplacian spectrum for digraphs. We also illustrate this Laplacian spectral excess theorem for digraphs with few Laplacian eigenvalues and we show that any strongly connected and regular digraph that has normal Laplacian matrix with three distinct eigenvalues, is distance-regular. Hence such a digraph is strongly regular with girth \(g=2\) or \(g=3\). MSC: 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05E30 Association schemes, strongly regular graphs 05C20 Directed graphs (digraphs), tournaments Keywords:Laplacian spectral excess theorem; distance-regular digraphs; strongly regular digraphs Citations:Zbl 0888.05056 PDFBibTeX XMLCite \textit{F. Shafiei}, Trans. Comb. 7, No. 3, 19--28 (2018; Zbl 1463.05354) Full Text: DOI References: [1] J. Bang-Jensen and G. Z. Gregory,Digraphs: Theory, Algorithms and Applications, Springer Monographs in Mathematics, 2nd ed, 2009. · Zbl 1170.05002 [2] A. E. Brouwer, personal homepage:http://www.cwi.nl/ aeb/math/dsrg/dsrg.html. [3] A. E. Brouwer, A. M. Cohen and A. Neumaier,distance-regular Graphs, Springer- Verlag, Berlin-New York, 1989. · Zbl 0747.05073 [4] A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Springer, 2012; available online athttp://homepages. cwi.nl/ aeb/math/ipm/. · Zbl 1231.05001 [5] F. Comellas, M. A. Fiol, J. Gimbert and M. Mitjana, Weakly distance-regular digraphs,J. Combin. Theory Ser. B, 90 (2004) 233-255. · Zbl 1033.05100 [6] D. M. Cvetkovic, M. Doob and H. Sachs,Spectra of Graphs, Theory and Application, VEB Deutscher Verlag der Wissenschaften, Berlin, second edition, 1982. · Zbl 0458.05042 [7] C. Dalfo, E.R. Van Dam, M. A. Fiol and E. Garriga, Dual concepts of almost distance-regularity and the spectral excess theorem,Discrete Math.,312(2012) 2730-2734. · Zbl 1245.05032 [8] C. Dalfo, E. R. Van Dam, M. A. Fiol, E. Garriga and B. L. Gorissen, On almost distance-regular graphs,J. Combin. Theory Ser. A,118(2011) 1094-1113. · Zbl 1225.05249 [9] R. M. Damerell, Distance-transitive and distance-regular digraphs,J. Combin. Theory Ser. B,31(1981) 46-53. · Zbl 0468.05034 [10] A. M. Duval, A directed graph version of strongly regular graphs,J. Combin. Theory Ser. A,47(1988) 71-100. · Zbl 0642.05025 [11] E. R. Van Dam, The spectral excess theorem for distance-regular graphs: a global (over)view,Electron. J. Combin.,15(2008), pp. 10. · Zbl 1180.05130 [12] E. R. Van Dam and M. A. Fiol, A short proof of the odd-girth theorem,Electron. J. Combin.,19(2012) pp. 5. · Zbl 1253.05098 [13] E. R. Van Dam and M. A. Fiol, The Laplacian Spectral Excess Theprem for distance-Regular Graphs,Linear Algebra Appl.,458(2014) 1-6. [14] Carl D. Meyer, Matrix analysis and applied linear algebra,Philadelphia, USA,101(2011) 486-489. [15] M. A. Fiol, Algebraic characterizations of distance-regular graphs,Discrete Math.,246(2002) 111-129. · Zbl 1025.05060 [16] M. A. Fiol, On some approaches to the spectral excess theorem for nonregular graphs,J. Combin. Theory Ser. A,120(2013) 1285-1290. · Zbl 1278.05089 [17] M. A. Fiol and E. Garriga, From local adjacency polynomials to locally pseudo-distance-regular graphs,J. Combin. Theory Ser. B,71(1997) 162-183. · Zbl 0888.05056 [18] M. A. Fiol, S. Gago and E. Garriga, A simple proof of the spectral excess theorem for distance-regular graphs, Linear Algebra Appl.,432(2010) 2418-2422. · Zbl 1221.05112 [19] M. A. Fiol, E. Garriga and J. L. A. Yebra, Locally pseudo-distance-regular graphs,J. Combin.Theory Ser. B, 68(1996) 179-205. · Zbl 0861.05064 [20] G. S. Lee and C. W. Weng, The spectral excess theorem for general graphs,J. Combin. Theory Ser. A,119 (2012) 1427-1431. · Zbl 1245.05087 [21] G. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.