Topological Integer Additive Set-Sequential Graphs

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by  Naduvath Sudev

Mathematics2015,3,604-614;doi:10.3390/math3030604
OPENACCESS
mathematics
ISSN2227-7390
www.mdpi.com/journal/mathematics
Article
Topological Integer Additive Set-Sequential Graphs
SudevNaduvath1,*,GerminaAugustine2 andChithraSudev3
1 DepartmentofMathematics,VidyaAcademyofScience&Technology,Thrissur-680501,Kerala,
6
1 India.
0
2 PG&ResearchDepartmentofMathematics,MaryMathaArts&ScienceCollege,Mananthavady-
2
b 670645,Kerala,India.
e
3 NaduvathMana,Nandikkara,Thrissur-680301,India.
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] *Authortowhomcorrespondenceshouldbeaddressed;e-mail:[email protected],telephone:
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[phone].
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.
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a Received: 7April2015/Accepted: 26June2015/Published: 3July2015
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[

1
v Abstract: LetN denotethesetofallnon-negativeintegersandX beanynon-emptysubset
0
5 of N . Denote the power set of X by P(X). An integer additive set-labeling (IASL) of
3 0
4 a graph G is an injective set-valued function f : V(G) → P(X) such that the induced
3
0 function f+ : E(G) → P(X) is defined by f+(uv) = f(u) +f(v), where f(u) +f(v) is
.
2 the sumset of f(u) and f(v). If the associated set-valued edge function f+ is also injective,
0
6 then such an IASL is called an integer additive set-indexer (IASI). An IASL f is said to be
1
a topological IASL (TIASL) if f(V(G))∪{∅} is a topology of the ground set X. An IASL
:
v
i is said to be an integer additive set-sequential labeling (IASSL) if f(V(G))∪f+(E(G)) =
X
P(X) − {∅}. An IASL of a given graph G is said to be a topological integer additive
r
a
set-sequential labeling of G, if it is a topological integer additive set-labeling as well as an
integer additive set-sequential labeling of G. In this paper, we study the conditions required
for a graph G to admit this type of IASL and propose some important characteristics of the
graphswhichadmitthistypeofIASLs.
Keywords: Integeradditiveset-labeling,integeradditiveset-sequentiallabeling,topological
integeradditiveset-labeling,topologicalintegeradditiveset-sequentiallabeling.
MSCclassifications: 05C78
Mathematics2015,3 605
1.Introduction
For all terms and definitions, other than newly defined or specifically mentioned in this paper, we
refer to [4] and [11] and for different graph classes, we further refer to [5] and [6]. For the terms and
concepts in Topology, we further refer to [9], [12] and [10]. Unless mentioned otherwise, the graphs
consideredinthispaperaresimple,finite,non-trivialandconnected.
Let A and B be two non-empty sets. The sumset of A and B is denoted by A+B and is defined by
A+B = {a+b : a ∈ A,b ∈ B}. If a set C = A+B, then A and B are said to the summands of C.
Usingtheconceptsofsumsetsoftwosets,thefollowingnotionhasbeenintroduced.
Let N denote the set of all non-negative integers. Let X be a non-empty subset of N and P(X)
0 0
be its power set. An integer additive set-labeling (IASL) of a graph G is an injective function f :
V(G) → P(X)suchthattheimageoftheinducedfunctionf+ : E(G) → P(N ),definedbyf+(uv) =
0
f(u)+f(v)is containedinP(X), wheref(u)+f(v)is thesumset off(u)andf(v)(see [7],[8]). It is
to be noted that for an IASL f, we have f+(E(G)) ⊆ P(X). A graph that admits an IASL is called an
integer additive set-labeled graph (IASL-graph). An IASL f of a given graph G is said to be an integer
additiveset-indexer(IASI)iftheassociatedfunctionf+ isalsoinjective.
If X = N , then it can be easily verified that every graph admits an integer additive set-labeling and
0
anintegeradditiveset-indexer.
The cardinality of the set-label of an element (a vertex or an edge) of a graph G is said to be the
set-indexingnumberofthatelement. Sincetheset-labelofeveryedgeofGisthesumsetoftheset-labels
of its end vertices, we do not permit labeling vertices and hence the edges of an IASL-graph G with the
emptyset. Ifanyofthegiventwosetsiscountablyinfinite,thentheirsumsetisalsoacountablyinfinite
set. Hence, all sets we consider in this paper are non-empty finite sets of non-negative integers. More
studiesondifferenttypesofintegeradditiveset-labeledgraphshavebeendonein[13–16].
An IASL f of a graph G, with respect to a ground set X ⊂ N , is said to be a topological IASL
0
(TIASL)ofGifT = f(V(G))∪{∅}isatopologyonX (see[17]). Certaincharacteristicsandstructural
propertiesofTIASL-graphshavebeenstudiedin[17].
ThefollowingarethemajorresultsontopologicalIASL-graphsobtainedin[17].
Proposition1. [17]Iff : V(G) → P(X)−{∅}isaTIASLofagraphG,thenGmusthaveatleastone
pendantvertex.
Proposition 2. [17] Let f : V(G) → P(X)−{∅} be a TIASL of a graph G. Then, the vertices whose
set-labels contain the maximal element of the ground set X are pendant vertices, adjacent to the vertex
havingtheset-label{0}.
If 0 ∈ X, then every subset A of X can be written as a sumset as A = {0} + A . This sumset
i i i
representationisknownasatrivialsumsetrepresentationofA andthesummands{0}andA oftheset
i i
A arecalledtrivialsummandsofA inX.
i i
Motivated by the studies of set-sequential graphs made in [1] and [2] and the studies of topological
set-sequentialgraphsmadein[3],weextendourstudiesonintegeradditiveset-sequentialgraphs,studied
in[19],forwhichthecollectionofset-labelsofwhoseelementsformsatopologyofthegroundsetX.
Mathematics2015,3 606
2.TopologicalIASS-Graphs
Letusfirstrecallthedefinitionofanintegeradditiveset-sequentiallabelingofagivengraphG.
Definition 3. [19] An integer additive set-labeling f of a graph G, with respect to a finite set X, is
said to be an integer additive set-sequential labeling (IASSL) of G if there exists an extension function
f∗ : G → P(X)off definedby

f(x) ifx ∈ V(G)
f∗(G) =
f+(x) ifx ∈ E(G)
suchthatf∗(G)∪{∅} = P(X).
A graph which admits an integer additive set-sequential labeling is called an integer additive
set-sequentialgraph(IASS-graph).
Weshallnowintroducethenotionofatopologicalintegeradditiveset-sequentialgraphsasfollows.
Definition 4. [19] An integer additive set-labeling f of a graph G, with respect to a finite set X, is said
to be a topological integer additive set-sequential labeling (TIASSL) if f(V)∪{∅} is a topology on X
andf∗(G) = P(X)−{∅},wheref∗(G) = f(V(G))∪f+(E(G)),asdefinedinthepreviousdefinition.
In other words, a TIASSL of a graph G is an IASI of G which is both a TIASL as well as an IASSL
ofG.
A topological integer additive set-sequential labeling f : V(G) → P(X) is said to be a topological
integeradditiveset-sequentialindexer(TIASSI)iftheinducedfunctionf∗ isalsoinjective.
Remark 1. Let f be a TIASSL of a given connected graph G, with respect to a given ground set X.
Then,sincef(V(G))∪{∅}isatopologyofX,X ∈ f(V). Therefore,{0} ∈ f(V)andthevertexhaving
the set-label X can be adjacent only to the vertex having {0}. Then, the vertex having the set label X is
apendantvertexofG.
An IASL f : V(G) → P(X), with respect to a finite ground set X, is said to be an integer
additive set-graceful labeling if the induced edge-function f+ : E(G) → P(X) satisfies the condition
f+(E(G)) = P(X)−{∅,{0}}(see[18]).
A topological integer additive set-graceful labeling (Top-IASGL) with respect to a finite ground set
X isdefinedin[20]asanintegeradditiveset-gracefullabelingf ofagraphGsuchthatf(V(G))∪{∅}
isatopologyonX (see[20]).
ThefollowingresultestablishestherelationbetweenaTIASGLandaTIASSLofG.
Theorem5. EveryTIASGLofagivengraphGisaTIASSLofG.
Proof. Let f be a TIASGL of a given graph G. Then, f(V) ∪ {∅} is a topology on X and f+(E) =
P(X)−{∅,{0}}. Also,wehavef(V(G))−f+(E(G)) = {0}andf(V)−{0} ⊆ f+(E(G)). Therefore,
f(V)∪f+(E) = P(X)−{∅}. Therefore,f isaTIASSLofagraphG.
Mathematics2015,3 607
The converse of this result need not be true. For, the fact f(V) ∪ f+(E) = P(X) − {∅} does not
necessarily imply that f+(E) = P(X) − {∅,{0}}. Figure 1 is an example of a TIASS-graph which
is not a TIASG-graph with respect to the ground set X. Indeed, the sets {1,2} and {0,2} are not the
set-labelsofanyedgeofthegraphandhencewecanseethatthesubsetsf+(E) (cid:54)= P(X)−{∅,{0}}.
Figure1. AnexampleofanTIASS-graphwhichisnotaTIASG-graph
By a graphical realisation of a collection P(X) of subsets of a set X ⊆ N with respect to an IASL
0
f, we mean a graph G such that all non-empty elements of P(X) are the set-labels of some elements
(edgesorvertices)ofG.
In view of the above notions and concepts, the following theorem establishes the necessary and
sufficient condition for a graph to admit a TIASSL with respect to the discrete topology of a given
groundsetX.
Theorem 6. A graph G admits a TIASSL f with respect to the discrete topology of a non-empty finite
∼
setX ifandonlyifG = K .
1,2|X|−2
∼
Proof. LetG = K . Then,GcanbeagraphicalrealisationofthecollectionP(X)−∅. Therefore,
1,2|X|−2
thecorrespondingIASIf isaTop-IASGLofG. Then,byTheorem5,f isaTIASSL.
Conversely, assume that G admits a TIASSL, say f, with respect to the discrete topology of a
non-empty finite set X. Therefore, |f(V(G))| = 2|X| − 1 and |f+(E(G))| = 2|X| − 2. Therefore,
G is a tree on 2|X| − 1 vertices. Moreover, f(V(G)) = f+(E(G)) ∪ {0}, which possible only when
∼
G = K .
1,2|X|−2
Invokingtheabovetheorem,wehave
Theorem7. LetX beanon-emptyfinitesetofnon-negativeintegers,whichcontains0. Then,forastar
graphK ,theexistenceofaTop-IASGLisequivalenttotheexistenceofaTIASSL,withrespectto
1,2|X|−2
thegroundsetX.
Proof. Since 0 ∈ X, we have {0} ∈ P. Now, label the central vertex of K by {0} and label its
1,2|X|−2
end vertices by the other non-empty subsets ofX in an injective manner. Therefore, obviously, we have
f(V(G)) = P(X)−{∅} (and hence f∗(G) = P(X)−{∅}) and f+(E(G)) = P(X)−{∅,{0}}. This
completestheproof.
Mathematics2015,3 608
Now that the admissibility of a TIASSL by a graph with respect to the discrete topology of a given
non-empty ground set X has been discussed, we shall proceed to consider other topologies (containing
the set {0}) of X which are neither the discrete topology nor the indiscrete topology of X as the set of
the set-labels of vertices of G. In the following discussions, all the topologies we mention are neither
discretetoplogiesnorindiscretetoplogiesofX.
The question that arises first about the existence of a graph G that admits a TIASSL with respect to
anygivenfinitesetofnon-negativeintegers. Inthefollowingresultweestablishtheexistenceofagraph
whichadmitsaTIASLwithrespecttoagivengroundsethavingcertainproperties.
Theorem 8. Let X be a finite set of non-negative integers containing 0. Then, there exists a graph G
whichadmitsanTIASSLwithrespecttothegroundsetX.
Proof. Let X be a given non-empty finite set containing the element 0. Let us begin the construction
of the required graph as follows. First, choose a topology T on X such that the non-empty sets in T
togetherwiththeirsumsetsmakesthepowersetP(X).
Take a vertex v and label this vertex with the set {0}. Let A (cid:54)= {0} be a subset of X which is not
1 i
the non-trivial sumset of any two subsets of X. Then, create a new vertex v and label v by the set
i i
A . Join this vertex to v . Repeat this process until all the subsets of X which are not the non-trivial
i 1
sumsets of any subsets of X are used for labeling some vertices of the graph under construction. Since
f(V(G))∪{∅}mustbeatopologyonX,X mustbelongtof(V(G)). Hence,makeanadditionalvertex
v withtheset-labelX. Thisvertexmustalsobeadjacenttov .
1
NowthatallsubsetsofX whicharenotsumsetsofanytwosubsetsofX havebeenusedforlabeling
the vertices of the graph under construction the remaining subsets of X that are not used for labeling
are sumsets of some two subsets of X, both different from {0}. If A(cid:48) is a non-trivial sumset, then
r
we can choose its summands to be indecomposable as non-trivial sumsets set in P(X) such that A(cid:48) =
r
f(v )+f(v ),forsomeverticesv ,v ofthegraph. Hence,drawanedgee betweenv andv sothate
i j i j r i j r
has the set-label A(cid:48). Repeat this process until all other non-empty subsets of G are also the set-labels of
r
some edges in G. Clearly, under this labeling we get f∗(G) = P(X)−{∅}. Therefore, the graph G we
constructednowadmitsaTIASSLwithrespecttothegivengroundsetX.
Figure 2 depicts the existence of a TIASS-graph with respect to a given non-empty finite set of
non-negativeintegers.
In general, all graphs need not necessarily admit a TIASSL. It is evident from Proposition 1. In
this context, it is relevant and interesting to determine the structural properties of TIASS-graphs. The
minimumnumberofverticesrequiredinagraphtoadmitaTIASS-graphisdeterminedinthefollowing
result.
Proposition9. LetthegraphGadmitaTIASSLf withrespecttoagivengroundsetX andletAbethe
setofsubsetsofX,whicharenotnon-trivialsumsetsofanysubsetsofX. If|A| = ρ,thenGmusthave
(i) atleastρverticesifX ∈ A.
(ii) atleast1+ρverticesifX ∈/ A.
Mathematics2015,3 609
Figure2. ATIASS-graphwithrespecttothegroundsetX = {0,1,2,3}.
Proof. Since f is an IASI of G, the set-label of every edge of G is the sumset of two subsets of X.
Then, no element of A can be the set-label of any edge of G. Therefore, since f is an IASSL of G, all
theelementsofAmustbetheset-labelsofsomeverticesofG.
Since f is a TIASL also, X ∈ f(V). If X ∈ A, then G must have at least ρ vertices. If X ∈/ A,
then G must have one vertex corresponding to X and one vertex corresponding to each element of A.
Therefore,inthiscase,Gmusthaveatleast1+ρvertices.
In view of the above results, we note that the number of vertices adjacent to the vertex having the
set-label {0} is important in the study of TIASS-graphs. The following result describes the number of
verticesthatareadjacenttothevertexhavingtheset-label{0}.
Proposition 10. Let the graph G admit a TIASSL f with respect to a given ground set X. Then, G has
at least ρ(cid:48) vertices adjacent to the unique vertex of G, labeled by {0}, where ρ(cid:48) is the number of subsets
ofX whicharenotthesummandsofanysubsetsofX.
Proof. Let f be an IASSL of a given graph G with respect to the ground set X. Let B be the collection
of all subsets of X which are not the summands of any subsets of X. That is, no subset of X can be
expressed as a sumset of any two elements in B. Hence |B| = ρ(cid:48). Since no element of B is a summand
ofanysubsetofX,thennovertexinGwhichhastheset-labelfromB canbeadjacentonlytothevertex
of G having the set-label {0}. Therefore, the minimum number of vertices adjacent to the vertex with
set-label{0}isρ(cid:48).
Since f is a TIASL, by Remark 1, G must have some pendant vertices. The minimum number of
pendantverticesrequiredforagraphGtoadmitaTIASSLisdeterminedinthefollowingresult.
Mathematics2015,3 610
Proposition 11. Assume that the graph G admits a TIASSL f with respect to a given ground set X and
let D be the set of subsets of X which are neither the non-trivial sumsets of two subsets of X nor the
summandsofanysubsetsofX. If|D| = ρ(cid:48)(cid:48),thenGmusthave
(i) atleastρ(cid:48)(cid:48) pendantverticesifX ∈ D.
(ii) atleast1+ρ(cid:48)(cid:48) pendantverticesifX ∈/ D.
Proof. ByProposition9,noelementofD canbetheset-labelofanyedgeofG. Also,sincenoelement
ofD isasummandofanysubsetofX,novertexinGthathastheset-labelfromofD canbeadjacentto
anyothervertexinGotherthantheonehavingtheset-label{0}. Therefore,noelementofD canbethe
set-labelofanelementofGthatisnotapendantvertexofG.
Clearly,X isnotasummandofanysubsetofX. Then,X ∈ D ifandonlyX isnotthesumsetofany
subsetsofX. Therefore,ifX ∈ D,thentheminimumnumberofpendantverticesinGisatleastρ(cid:48)(cid:48).
If X ∈/ D, then X is a non-trivial sumset of two subsets of X. But, since f is a TIASL, X ∈ f(V).
Therefore, the vertex of G whose set-label is X, can be adjacent only to the vertex of G having the
set-label {0}. Therefore, in addition to the vertices with set-labels from D, the vertex with set-label X
isalsoapendantvertex. Hence,inthiscase,thenumberofpendantverticesinGisatleast1+ρ(cid:48)(cid:48).
The number of edges in the given graph G is also important in determining the admissibility of a
TIASSL by G. Since the minimum number of vertices required in a TIASS-graph have already been
determined,wecandeterminethemaximumnumberofedgesrequiredinG.
Invokingtheaboveresultsandconcepts,weproposethefollowingtheorem.
Proposition 12. Let the graph G admit a TIASSL f with respect to a given ground set X. Let A be the
set of subsets of X, which are not the sumsets of any subsets of X and B be the collection of all subsets
of X which are not the summands of any subsets of X. Also, let |A| = ρ and |B| = ρ(cid:48). Then, G has at
most2|X| +ρ(cid:48) −ρ−1edges.
Proof. By Proposition 10, every element of B is the set-label of a vertex of G which is adjacent to the
vertexhavingthe set-label{0}. Therefore,everyelementB ofB istheset-label ofonevertexofGand
i
theedgebetweenthatvertexandthevertexwiththeset-label{0}. Sincetheset-labelofeveryedgeisthe
sumsetoftheset-labelsofitsendvertices,themaximumnumberofremainingedgesinGisthenumber
ofsubsetsofX whicharethesumsetstwosubsetsofX.
Case-1: Let X ∈/ A. Recall that X ∈ B. Then, the maximum number of edges is equal to the number
of elements in the set A(cid:48), which is the set of all (non-empty) subsets of X which are non-trivial sumsets
of two subsets of X. We have A(cid:48) = P(X)−(A∪{0}). Hence, maximum number of remaining edges
in G is 2|X| −(|A|+1) = 2|X| −ρ−1. Therefore, in this case, the maximum number of edges in G is
2|X| +ρ(cid:48) −ρ−1.
Case-2: Let X ∈ A. Then, by Proposition 10, G has at least 1+ρ(cid:48) vertices adjacent to a unique vertex,
having set-label{0}. Therefore, there are at least 1 + ρ(cid:48) edges in G with set-labels from B ∪ X. The
maximum number of remaining edges is equal to the number of subsets of X, other than X, which
are the sumsets of any two subsets of X. Therefore, the maximum number of remaining edges in
Mathematics2015,3 611
G is 2|X| − |A| − 2 = 2|X| − ρ − 2. Hence, the maximum number of edges in G in this case is
1+ρ(cid:48) +2|X| −ρ−2 = 2|X| +ρ(cid:48) −ρ−1.
Letussummarisetheabovepropositionsinthefollowingtheorem.
Theorem 13. Let f be a topological integer additive set-sequential labeling defined on a graph with
respect to a given ground set X containing 0. Let A be the set of all subsets of X which are not the
non-trivialsumsetsofanytwosubsetsofX andBbethethesetofallsubsetsofX,whicharenon-trivial
summandsinsomesubsetsofX. Then,
(i) thenumberofverticesinGisatleast|A|or|A|+1inaccordancewithwhetherX isinAornot,
(ii) themaximumnumberofedgesinGis|B|+|A(cid:48)|,whereA(cid:48) = P(X)−(A∪{∅}),
(iii) theminimumdegreeofthevertexthathastheset-label{0}inGis|B|,
(iv) theminimumnumberofpendantverticesis|A∩B|or|A∩B|+1inaccordancewithwhetheror
notX isinA.
Now,letusverifywhetheragraphadmitsaTop-IASIwithrespecttoagivennon-emptyfinitesetX.
Proposition14. Noconnectedgraphadmitsatopologicalintegeradditiveset-sequentialindexer.
Proof. If possible, let f be a TIASSI defined on a given connected graph G. Then, f is a TIASL on G.
Then,X ∈ f(V). Then,thevertexv havingset-labelX mustbeapendantvertexanditmustbeadjacent
to the vertex u having the set-label {0}. Therefore, f∗(v) = f(v) = f+(uv) = f∗(uv), a contradiction
totheassumptionthatf isanTIASSIofG. Hence,f cannotbeaTIASSIofthegraphG.
Analogous to the Theorem 13, we propose the characterisation of a (disconnected) graph that admits
aTIASSIwithrespecttoagivengroundsetX.
Theorem 15. Let f be a topological integer additive set-sequential indexer defined on a graph with
respect to a given ground set X containing 0. Let A be the set of all subsets of X which are not the
non-trivialsumsetsofanytwosubsetsofX andBbethethesetofallsubsetsofX,whicharenon-trivial
summandsofanysubsetsofX. Then,
(i) the minimum number of vertices in G is |A| or |A|+1 in accordance with whether X is in A or
not,
(ii) theminimumnumberofisolatedverticesinGis|A∩B|or|A∩B|+1inaccordancewithwhether
X isinAornot,
(iii) themaximumnumberofedgesinGis|A(cid:48)|,whereA(cid:48) = P(X)−(A∪{∅}).
Proof. (i) TheproofissimilartotheproofofProposition9.
(ii) As we have proved earlier any vertex having the set-label from A ∩ B can be adjacent only to
the vertex having {0}. But, since no two elements of G can have the same set-label, no element
in A ∩ B can be adjacent to any other vertex of G. Therefore, the minimum number of isolated
verticesinGis|A∩B|.
Mathematics2015,3 612
(iii) Since all elements in A ∩ B corresponds to isolated vertices in G, the number of non-isolated
verticesofGis|A−B|. Therefore,allelementsinA(cid:48) arethesumsetsoftheelementsin|A−B|.
Therefore,themaximumnumberedgesinGisequalto|A(cid:48)| = 2|X| −ρ−1.
Thiscompletestheproof.
3.Conclusion
In this paper, we have discussed certain properties and characteristics of topological IASL-graphs.
MorepropertiesandcharacteristicsofTIASLs,bothuniformandnon-uniform,areyettobeinvestigated.
Thefollowingaresomeproblemswhichrequirefurtherinvestigation.
Problem 16. Determine the smallest ground set X, with respect to which different graph classes admit
TIASSLs.
Problem17. CharacterisethegraphswhichareTIASS-graphs,butnotTIASG-graphs.
Problem18. CheckwhethertheconversesofTheorem13andTheorem15aretrue.
Problem 19. Check the admissibility of TIASSL by different operations and products of given
TIASS-graphs.
The problems of establishing the necessary and sufficient conditions for various graphs and graph
classes to have certain other types IASLs are also still open. Studies of those IASLs which assign sets
havingspecificproperties,totheelementsofagivengrapharealsonoteworthy.
Acknowledgments
Theauthorswouldliketothanktheanonymousreviewersfortheirinsightfulsuggestionsandcritical
andconstructiveremarkswhichmadetheoverallpresentationofthispaperbetter.
AuthorContributions
Allthreeauthorshavesignificantcontributiontothispaperandthefinalformofthispaperisapproved
byallthreeauthors.
ConflictsofInterest
Theauthorsdeclarethattheyhavenoconflictofinterestsregardingthepublicationofthearticle.
References
1. Acharya, B. D.; Germina, K. A.; Abhishek, K.; Slater, P. J. Some New Results on Set-Graceful and
Set-SequentialGraphs. JournalofCombinatorics,InformationandSystemSciences,2012,37,(2-4),
145-155.
Mathematics2015,3 613
2. Acharya, B. D.; Hegde, S. M. Set-Sequential Graphs. National Academy of Science Letters,
1985,8(12),387-390.
3. Acharya, B. D.; Germina, K. A.; Princy, K. L.; Rao, S. B. Embedding and NP-completeness
of set-graceful, topologically set-graceful and set-sequentaial graphs. Journal of Discrete
MathematicalSciences&Cryptography,2009,12(2),481-487.
4. Bondy,J.A.;Murty,U.S.R.GraphTheory. Springer,2008.
5. Brandstädt,A.;Le,V.B.;Spinrad,J.P.fGraphClasses: ASurvey. SIAM,Philadelphia,1987.
6. Gallian, J. A. A Dynamic Survey of Graph Labelling. The Electronic Journal of Combinatorics, (#
DS-6),2013.
7. Germina, K. A.; Anandavally, T. M. K. Integer Additive Set-Indexers of a Graph: Sum Square
Graphs. JournalofCombinatorics,InformationandSystemSciences,2012,37(2-4),345-358.
8. Germina, K. A.; Sudev, N. K. On Weakly Uniform Integer Additive Set-Indexers of Graphs.
InternationalMathematicalForum,2013,8(37),1827-1834.,DOI:10.12988/imf.2013.310188
9. Joshy,K.D.IntroductiontoGeneralTopology. NewAgeInternational,1983.
10. Krishnamoorthy, V. On the Number of Topologies of Finite Sets. The American Mathematical
Monthly,1966,73(2),154-157.
11. Harary,F.GraphTheory. Addison-WesleyPublishingCompanyInc.,1969.
12. Munkers,J.R.Topology. PrenticeHall,2000.
13. Sudev,N.K.;Germina,K.A.OnIntegerAdditiveSet-IndexersofGraphs. InternationalJournalof
MathematicalSciences&EngineeringApplications,2014,8(2),11-22.
14. Sudev, N. K.; Germina, K. A. Some New Results on Strong Integer Additive Set-Indexers
of Graphs. Discrete Mathematics, Algorithms & Applications, 2015,7(1), 1-11., DOI:
10.1142/S1793830914500657.
15. Sudev, N. K.; Germina, K. A.; Chithra, K. P. A Creative Review on Integer Additive Set-Labeled
Graphs.Asian-EuropeanJournalofMathematics,(2015),toappear.
16. Sudev,N.K.;Germina,K.A.TheexquisiteIntegerAdditiveSet-LabelingofGraphs. International
JournalofScienceandResearch,2015,4(3),2858-2862.
17. Sudev, N. K.; Germina, K. A. A Study on Topological Integer Additive Set-Labeling of
Graphs. Electronic Journal of Graph Theory and Applications, 2015,3(1), 70âA˘S¸84, DOI:
10.5614/ejgta.2015.3.1.8.
18. Sudev,N.K.;Germina,K.A.AStudyonIntegerAdditiveSet-GracefulGraphs. underreview.
19. Sudev,N.K.;Germina,K.A.OnIntegerAdditiveSet-SequentialGraphs. underreview.
20. Sudev,N.K.;Germina,K.A.TopologicalIntegerAdditiveSet-GracefulGraphs. underreview.
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