Partial Actions on Graphs
R.P. Sharma1, Rajni Parmar1, V.S. Kapil2
1Department of Mathematics, Himachal Pradesh University, Shimla-171005, India
2Department of Mathematics, Govt. College Jukhala, Bilaspur-174033, India
*Corresponding Author Email: rp_math_hpu@yahoo.com, vik_math@yahoo.com
ABSTRACT:
We define a
partial action of a group on a graph and their partial orbits and stabilizers
for partial graphs. Further some relations between partial orbits and
stabilizers are proved. A relation between 
transitivity and
transitivity of
a partial graph is proved where α is a partial action of a group
on the set of
vertices and β is a partial action of a group 
on the edge set
of the graph.
KEYWORD:
1. INTRODUCTION.
Graph Theory is a branch of Mathematics
for the exploration of techniques in Discrete Mathematics and the results of
graph theory are applied in many areas of the computing, social and natural
sciences. The interaction between group theory (see [3]) and topology occurs
where a group acts on a graph and information is obtained about the group or
about the graph, whenever these occur. The actions of groups are studied on
graphs by Warren Dicks and M.J. Dunwoody in [2]. The partial group action is a
generalization of group action on a given structure (see [4], [5], [6] and [7]).
Therefore, it is a very desirable feature to examine the partial actions of
groups on graphs. In this paper, we define a partial action of a group on a
graph, orbit and stabilizer for partial graphs and prove that
, the stabilizer
of 
in is a subgroup
of . Finally, we
find a relation between transitivity and
transitivity of
a partial graph.
REFERENCES:
1. P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, Basic Abstract Algebra, Cambridge University Press (1995).
2. W. Dicks and M.J. Dunwoody, Group acting on graphs, Cambridge University Press 1989.
3. R. Diestel, Graph theory, 1997 Springer-verlag New York, Inc.
4. M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions, enveloping actions and partial representations, Trans. Amer. Math. Society, 357(5)(2005), 1931-1952.
5. R.P. Sharma and Anu, Going up and Going Down Relations for partial Actions on Algebras, Research Journal of Science and Technology, 5(1)(2013), 144-147.
6. R.P. Sharma and Anu and N. Singh, Partial Group Actions on Semialgebras, Asian European Journal of Mathematics, 5(4)(2012), 1250060 (1-20).
7. R.P. Sharma and Meenakshi, Morita Equivalence between Partial Actions and Global Actions for Semialgebras, Journal of Combinations & System Sciences(JCISS) vol. 41 (2016).
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Received on 28.08.2016 Accepted on 14.09.2016 © EnggResearch.net All Right Reserved Int. J. Tech. 2016; 6(2): 227-232. DOI: 10.5958/2231-3915.2016.00035.3 |
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