Regularity of binomial edge ideals of certain block graphs
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We prove that the regularity of binomial edge ideals of graphs obtained by gluing two graphs at a free vertex is the sum of the regularity of individual graphs. As a consequence, we generalize certain results of Zafar and Zahid (Electron J Comb 20(4), 2013). We obtain an improved lower bound for the regularity of trees. Further, we characterize trees which attain the lower bound. We prove an upper bound for the regularity of certain subclass of block-graphs. As a consequence, we obtain sharp upper and lower bounds for a class of trees called lobsters.
KeywordsBinomial edge ideals Castelnuovo–Mumford regularity block graph tree
Mathematics Subject Classification13D02 05E40
The authors would like to thank Nathann Cohen for setting up SAGE and giving them initial lessons in programming. The authors have extensively used computer algebra software SAGE , and Macaulay2 , for their computations. Thanks are also due to Jinu Mary Jameson who provided the authors with a lot of computational materials. This research was partly funded by ICSR Exploratory Project (Grant MAT/1415/831/RFER/AVJA) of IIT Madras and Extra Mural Research Project by Sciences and Engineering Research Board, Government of India (Grant EMR/2016/001883). They would also like to thank the anonymous referee for a meticulous reading and making several suggestions which improved the exposition.
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