On Integer Additive Set-Valuations of Finite Jaco Graphs
DOI:
https://doi.org/10.26713/jims.v8i2.408Keywords:
Integer additive set-labeled graphs, Weak integer additive set-labeled graphs, Arithmetic integer additive set-labeled graphs, Dispensing number of a graph, Finite linear Jaco graphAbstract
Let \(X\) denote a set of non-negative integers and \(\mathcal{P}(X)\) be its power set. An integer additive set-labeling (IASL) of a graph \(G\) is an injective set-valued function \(f:V(G)\to \mathcal{P}(X)-\{\emptyset\}\) where induced function \(f^+:E(G) \to \mathcal{P}(X)-\{\emptyset\}\) is defined by \(f^+ (uv) = f(u)+ f(v)\), where \(f(u)+f(v)\) is the sumset of \(f(u)\) and \(f(v)\). Let \(f(x)=mx+c\); \(m\in \mathbb{N}\), \(c\in N_0\). A finite linear Jaco graph, denoted by \(J_n(f(x))\), is a directed graph with vertex set \(\{v_i: i\in \mathbb{N}\}\) such that \((v_i,v_j)\) is an arc of \(J_n(f(x))\) if and only if \(f(i)+i-d^-(v_j)\ge j\). In this paper, we discuss the admissibility of different types of integer additive set-labeling by finite linear Jaco graphs.Downloads
References
B.D. Acharya, Set-valuations and their applications, MRI Lecture Notes in Applied Mathematics, No. 2, The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad (1983).
J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, North-Holland, New York (1976).
N. Deo, Graph Theory with Application to Engineering and Computer Science, Prentice Hall of India, Delhi (1974).
K.A. Germina and N.K. Sudev, On weakly uniform integer additive set-indexers of graphs, Int. Math. Forum 8 (37) (2013), 1827–1834. doi: 10.12988/imf.2013.310188.
F. Harary, Graph Theory, Addison-Wesley (1969).
J. Kok, C. Susanth and S.J. Kalayathankal, A study on linear Jaco graphs, J. Inform. Math. Sci. 7 (2) (2015), 69–80.
J. Kok, P. Fisher, B. Wilkens, M. Mabula and V. Mukungunugwa, Characteristics of Jaco graphs, (J_1(a)), (a inmathbb{N}), arXiv: 1404.1714v1.
J. Kok, P. Fisher, B. Wilkens, M. Mabula and V. Mukungunugwa, Characteristics of finite Jaco graphs, (J_n(1)), (n inmathbb{N}), arXiv: 1404.0484v1.
M.B. Nathanson, Additive Number Theory, Inverse Problems and Geometry of Sumsets, Springer, New York (1996).
N.K. Sudev and K.A. Germina, On integer additive set-indexers of graphs, Int. J. Math. Sci. Eng. Appl. 8 (2) (2015), 11–22.
N.K. Sudev and K.A. Germina, A characterisation of weak integer additive set-indexers of graphs, J. Fuzzy Set Valued Anal. 2014 (2014), 1–6, doi: 10.5899/2014/jfsva-00189.
N.K. Sudev and K.A. Germina, Weak integer additive set-indexers of graph products, J. Inform. Math. Sci. 6 (1) (2014), 35-43.
N.K. Sudev and K.A. Germina, A note on the sparing number of graphs, Adv. Appl. Discrete Math. 14 (1) (2014), 51–65.
N.K. Sudev and K.A. Germina, A characterisation of strong integer additive set-indexers of graphs, Commun. Math. Appl. 5 (3) (2014), 101–110.
N.K. Sudev and K.A. Germina, Some new results on strong integer additive set-indexers of graphs, Discrete Math. Algorithms Appl. 7 (1) (2015), 1–11, doi: 10.1142/S1793830914500657.
N.K. Sudev and K.A. Germina, On certain arithmetic integer additive set-indexers of graphs, Discrete Math. Algorithms Appl. 7 (3) (2015), 1–15, doi: 10.1142/S1793830915500251.
N.K. Sudev and K.A. Germina, A study on topological integer additive set-labeling of graphs, Electron. J. of Graph Theory Appl. 3 (1) (2015), 70–84., doi: 10.5614/ejgta.2015.3.1.8.
N.K. Sudev and K.A. Germina, On integer additive set-sequential graphs, Int. J. of Math. Combin. 3 (2015), 125–133.
N.K. Sudev and K.A. Germina, Some new results on weak integer additive set-labelings of graphs, Int. J. Computer Appl. 128 (5) (2015), 1–5, doi: 10.5120/ijca2015906514.
N.K. Sudev and K.A. Germina, A study on topogenic integer additive set-labeled graphs, J. Adv. Res. Pure Math. 7 (3), 15–22, doi: 10.5373/jarpm.2230.121314.
N.K. Sudev and K.A. Germina, A study on arithmetic integer additive set-indexers of graphs, J. Adv. Res. Appl. Math. 8 (2) (2016), in press.
N.K. Sudev and K.A. Germina, A study on integer additive set-graceful graphs, J. Adv. Res. Pure Math. 8 (2) (2016), in press.
N.K. Sudev, K.P. Chithra and K.A. Germina, Integer additive set-filter graphs, Electron. J. Graph Theory Appl., to appear.
N.K. Sudev and K.A. Germina, A study on prime arithmetic integer additive set-indexers of graphs, communicated.
D.B. West, Introduction to Graph Theory, Pearson Education Inc. (2001).
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
