The endomorphisms monoids of graphs of order $n$ with a minimum degree $n-3$
Algebra and discrete mathematics, Tome 18 (2014) no. 2, pp. 274-294.

Voir la notice de l'article provenant de la source Math-Net.Ru

We characterize the endomorphism monoids, $\operatorname{End}(G)$, of the generalized graphs $G$ of order $n$ with a minimum degree $n-3$. Criteria for regularity, orthodoxy and complete regularity of those monoids based on the structure of $G$ are given.
Keywords: graph of order $n$ which minimal degree $n-3$, regular, completely regular.
Mots-clés : graph endomorphism, orthodox
@article{ADM_2014_18_2_a8,
     author = {Nirutt Pipattanajinda and Ulrich Knauer and Boyko Gyurov and Sayan Panma},
     title = {The endomorphisms monoids of graphs of order $n$ with a minimum degree $n-3$},
     journal = {Algebra and discrete mathematics},
     pages = {274--294},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2014_18_2_a8/}
}
TY  - JOUR
AU  - Nirutt Pipattanajinda
AU  - Ulrich Knauer
AU  - Boyko Gyurov
AU  - Sayan Panma
TI  - The endomorphisms monoids of graphs of order $n$ with a minimum degree $n-3$
JO  - Algebra and discrete mathematics
PY  - 2014
SP  - 274
EP  - 294
VL  - 18
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2014_18_2_a8/
LA  - en
ID  - ADM_2014_18_2_a8
ER  - 
%0 Journal Article
%A Nirutt Pipattanajinda
%A Ulrich Knauer
%A Boyko Gyurov
%A Sayan Panma
%T The endomorphisms monoids of graphs of order $n$ with a minimum degree $n-3$
%J Algebra and discrete mathematics
%D 2014
%P 274-294
%V 18
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2014_18_2_a8/
%G en
%F ADM_2014_18_2_a8
Nirutt Pipattanajinda; Ulrich Knauer; Boyko Gyurov; Sayan Panma. The endomorphisms monoids of graphs of order $n$ with a minimum degree $n-3$. Algebra and discrete mathematics, Tome 18 (2014) no. 2, pp. 274-294. http://geodesic.mathdoc.fr/item/ADM_2014_18_2_a8/

[1] D. Amar, “Irregularity strength of regular graphs of large degree”, Discrete Math., 114 (1993), 9–17 | DOI | MR | Zbl

[2] S. Fan, “On end-regular graphs”, Discrete Math., 159 (1996), 95–102 | DOI | MR | Zbl

[3] S. Fan, “Retractions of split graphs and End-orthodox split graphs”, Discrete Math., 257 (2002), 161–164 | DOI | MR | Zbl

[4] H. Hou, Y. Luo, “Graphs whose endomorphism monoids are regular”, Discrete Math., 308 (2008), 3888–3896 | DOI | MR | Zbl

[5] H. Hou, Y. Luo, Z. Cheng, “The endomorphism monoid of $\overline{P}_n$”, European J. Combinatorics, 29 (2008), 1173–1185 | DOI | MR | Zbl

[6] U. Knauer, “Endomorphisms of graphs II. Various unretractive graphs”, Arch. Math., 55 (1990), 193–203 | DOI | MR | Zbl

[7] U. Knauer, “Unretractive and $S$-unretractive joins and lexicographic products of graphs”, J. Graph Theory, 11:3 (1987), 429–440 | DOI | MR | Zbl

[8] U. Knauer, M. Nieporte, “Endomorphisms of graphs I. The monoid of strong endomorphisms”, Arch. Math., 52 (1989), 607–614 | DOI | MR | Zbl

[9] W. Li, “A regular endomorphism of a graph and its inverses”, Mathematika, 41 (1994), 189–198 | DOI | MR | Zbl

[10] W. Li, “Graphs with regular monoids”, Discrete Math., 265 (2003), 105–118 | DOI | MR | Zbl

[11] W. Li, “Green's relations on the endomorphism monoid of a graph”, Mathematica Slovaca, 45:4 (1995), 335–347 | MR | Zbl

[12] W. Li, “Split graphs with completely regular endomorphism monoids”, Journal of Mathematical Research and Exposition, 26:2 (2006), 253–263 | MR | Zbl

[13] W. Li, J. Chen, “Endomorphism-regularity of split graphs”, Europ. J. Combinatorics, 22 (2001), 207–216 | DOI | MR | Zbl

[14] N. Pipattanajinda, Sr. Arworn, “Endo-regularrity of cycle book graphs”, Thai Journal of Mathematics, 8:3 (2010), 99–104 | MR | Zbl

[15] N. Pipattanajinda, B. Gyurov, S. Panma, “Path strong and cycle strong graph endomorphism and applications”, Advances and Applications in Discrete Mathematics, 9:1 (2012), 29–44 | MR | Zbl

[16] N. Pipattanajinda, U. Knauer, Sr. Arworn, “Endo-Regularrity of Generalized Wheel Graphs”, Chamchuri Journal of Mathematics, 3 (2011), 45–57 | MR | Zbl

[17] N. Pipattanajinda, U. Knauer, B. Gyurov, S. Panma, “The endomorphism monoids of ($n-3$)-regular graphs of order $n$”, Algebra and Discrete Mathematics (to appear)

[18] M. Petrich and N. R. Reilly, Completely reguglar semigroups, Wiley Interscience, 1999 | MR

[19] J. Thomkeaw, Sr. Arworn, “Endomorphism monoid of $C_{2n+1}$ book graphs”, Thai Journal of Mathematics, 7:2 (2009), 319–327 | MR | Zbl

[20] W. Wang, H. Hou, “The endomorphism monoid of $N$-prism”, International Mathematical Forum, 50:6 (2011), 2461–2471 | MR

[21] A. Wanichsombat, “Endo-completely-regular split graphs”, Semigroups, Acts and Categories with Applications to Graphs, Proceedings (Tartu), 2007, 136–142 | MR

[22] A. Wilkeit, “Graphs with regular endomorphism monoid”, Arch. Math., 66 (1996), 344–352 | DOI | MR | Zbl