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for all terms and definitions , not defined specifically in this paper , we refer to @xcite . unless mentioned otherwise , all graphs considered here are simple , finite and have no isolated vertices . the researches in the field of graph labeling commenced with the introduction of the notion of graceful labeling of graphs in @xcite . analogous to the number valuations of graphs , the notion of set - valuations of graphs was introduced in @xcite . let us first consider the following notions of a set - labeling and a set - indexer of a given graph @xmath1 . [ d0 ]    [ d1 ] . note that if either @xmath8 or @xmath9 is countably infinite , then their sum set will also be countably infinite . hence , all sets we consider here are finite sets . we denote the cardinality of a set @xmath8 by @xmath10 . using the concepts of sum sets of finite sets , the notion of an integer additive set - indexer of a given graph @xmath1 has been introduced as follows . [ d2 ] . [ d3 ]    [ du ] @xcite an iasi is said to be _ @xmath5-uniform _ if @xmath6 for all @xmath7 . that is , a connected graph @xmath1 is said to have a @xmath5-uniform iasi if all of its edges have the same set - indexing number @xmath5 . in particular , we say that a graph @xmath1 has an _ arbitrarily @xmath5-uniform iasi _ if @xmath1 has a @xmath5-uniform iasi for every positive integer @xmath5 .    in @xcite , we have proved the following results on weakly uniform and arbitrarily uniform iasis . [ l1 ] @xcite let @xmath8 and @xmath9 be two non - empty finite subsets of @xmath0 . then @xmath11 . [ r1 ]    the following results are a necessary and sufficient condition for an iasi to be a weak iasi and a weakly uniform iasi respectively , of a given graph @xmath1 . [ l2a ] @xcite an iasi @xmath12 of a graph @xmath1 is a weak iasi if and only if at least one end vertex of every edge of @xmath1 has the set - indexing number @xmath13 . [ l2b ] @xcite an iasi @xmath12 of a graph @xmath1 is a weakly @xmath5-uniform iasi if and only if one end vertex of every edge of @xmath1 has the set - indexing number @xmath13 and the other end vertex has the set - indexing number @xmath5 , where @xmath5 is a positive integer .    in this paper , we study a special type of integer additive set - indexer of a graph @xmath1 called a strong integer additive set - indexer of @xmath1 . first note that all the sets we consider during this discussion , are non - empty , finite sets of non - negative integers . in view of lemma [ l1 ] , the maximum cardinality of the sum set of two non - empty finite sets is the product of the cardinalities of each individual sets . that is , for two non - empty finite sets @xmath8 and @xmath9 , we have @xmath14 . in this context , the study regarding characteristics of graphs in which the set - indexing number of every edge is the product of the set - indexing numbers of its end vertices arises much interest . hence , we define    [ d6 ]    [ t2 ] let @xmath8 , @xmath9 be two non - empty distinct subsets of @xmath0 . then , @xmath15 if and only if the differences between any two elements of one set are not equal to any differences between any two elements of the other .    due to lemma [ l1 ] , we have @xmath16 . we have @xmath17 if and only if @xmath18 for some @xmath19 and @xmath20 . that is , when @xmath21 . therefore , @xmath15 if and only if @xmath22 for any @xmath19 and @xmath20 . let @xmath23 and let @xmath24 . then , it is clear that @xmath25 if and only if no element of @xmath26 is equal to an element of @xmath27 . the set of all positive differences between any two elements of a set @xmath8 is called the _ difference set _ of @xmath8 and can be denoted by @xmath26 . that is , @xmath28 . hence , we propose the following theorem which establishes a necessary and sufficient condition for a complete graph @xmath29 to admit a strong iasi . let @xmath8 and @xmath9 be two non - empty subsets of @xmath0 . we use the notation @xmath30 in the sense that @xmath31 for all @xmath32 and @xmath33 . that is , @xmath34 . also , by the sequence @xmath35 , we mean that all the sets @xmath36 are pairwise disjoint . [ tsk ] let each vertex @xmath37 of the complete graph @xmath29 be labeled by the set @xmath38 . then , @xmath29 admits a strong iasi if and only if there exists a finite sequence of sets @xmath39 where each @xmath40 is the difference set of the set - label @xmath36 . assume that @xmath29 admits a strong iasi . let @xmath41 be the vertex set of @xmath29 . let each vertex @xmath37 is labeled by the set @xmath36 in @xmath42 . define the set @xmath40 by @xmath43 which is the set of all differences between any two elements of @xmath36 . since the vertex @xmath44 is adjacent to @xmath45 , without loss of generality , assign the set @xmath46 to @xmath44 and the set @xmath47 to the vertex @xmath45 . then , by lemma [ t2 ] , we have @xmath48 . since @xmath45 is adjacent to @xmath49 , assign @xmath50 to @xmath49 so that @xmath51 . combining these two , we get @xmath52 . proceeding in this way , for @xmath53 , assign the set @xmath36 to @xmath37 and the set @xmath54 to @xmath55 so that @xmath56 . after all possible assignments , we get @xmath39 . conversely , assume that each vertex @xmath37 of the complete graph @xmath29 is labeled by a non - empty set @xmath38 such that there exists a finite sequence of sets @xmath39 where each @xmath40 is the set of all differences between any two elements of the set @xmath36 . then , for each edge @xmath57 we have @xmath56 . hence , by lemma [ t2 ] , @xmath58 for all @xmath59 . therefore , @xmath1 admits a strong iasi . [ tss ] if a graph @xmath1 admits a strong iasi , then its subgraphs also admit strong iasi . the following result is a general condition for a general graph to admit a strong iasi . [ tsc ] a connected graph @xmath1 ( on @xmath60 vertices ) admits a strong iasi if and only if each vertex @xmath37 of @xmath1 is labeled by a set @xmath36 in @xmath42 and there exists a finite sequence of sets @xmath61 , where @xmath62 is a positive integer and each @xmath40 is the set of all differences between any two elements of @xmath36 .    in view of theorem [ tss ] , an interesting question we need to answer is whether a strong iasi of a graph @xmath1 induces to some other graph structures formed from @xmath1 by removing or identifying some of its elements ( vertices or edges ) . first , recall the following notion . the following theorem establishes a necessary and sufficient condition for a graph homeomorphic to a given strong iasi graph @xmath1 to admit a strong iasi . let @xmath1 be a graph which admits a strong iasi . then , any graph @xmath63 , obtained by applying a finite number of elementary topological reductions on @xmath1 , admits a strong iasi if and only if there exist the same number of paths @xmath64 , not the part of any triangle in @xmath1 , such that the difference sets of the set - labels of their vertices are pairwise disjoint . let @xmath1 be a graph which admits a strong iasi . let @xmath65 be a vertex of @xmath1 with @xmath66 . then @xmath65 is adjacent two vertices @xmath67 and @xmath68 in @xmath1 . then we have @xmath69 and @xmath70 . let @xmath71 , @xmath72 and @xmath73 be the labeling sets of @xmath74 respectively . also let @xmath75 be the corresponding sets of all differences between the elements of @xmath76 respectively . assume that @xmath77 admits a strong iasi . then , @xmath78 . therefore , by [ t2 ] , the sets all differences of the set - labels of @xmath67 and @xmath68 are pairwise disjoint in @xmath79 . that is , @xmath80 in @xmath79 . hence , the path @xmath81 in @xmath1 whose vertices have the set - labels with the difference sets of these vertices are pairwise disjoint . conversely , assume that the difference sets of the set - labels of whose vertices are pairwise disjoint.that is , @xmath82 . now , delete @xmath65 from @xmath1 . let @xmath83 . here , we have @xmath80 . then , by lemma [ t2 ] , @xmath78 in @xmath79 . all other vertices and edges are the same in @xmath79 and @xmath1 . therefore , @xmath79 admits a strong iasi . this completes the proof . figure [ ghom ] depicts the existence of strong iasis of the graphs @xmath84 , and @xmath85 which are obtained by applying finite number of elementary topological reductions on @xmath1 .        in the following section , we are going to discuss about the strongly uniform integer additive set - indexers of given graphs . in other words , we can say that a strong integer additive set - indexer @xmath12 of a graph @xmath1 is said to be a _ strongly @xmath5-uniform integer additive set - indexer _ if @xmath86 for all @xmath7 , where @xmath5 is a positive integer . in view of the above definition , we note the following . [ r6 ]    [ r7 ]    let us now proceed to discuss the admissibility of strongly uniform iasis by certain graphs and conditions required for their existence for various graphs . the following theorem establishes the admissibility of strongly uniform iasis by bipartite graphs . [ tbs ] for any positive integer @xmath5 , all bipartite graphs admit a strongly @xmath5-uniform iasi . the following result is a special case when the number @xmath5 , we consider above is a prime number . [ cbp ] if @xmath87 is a prime number , then a strongly @xmath87-uniform iasi of a bipartite graph @xmath1 is also a weakly @xmath87-uniform iasi for @xmath1 . invoking theorem [ tbs ] and corollary [ cbp ] , we have the following result . [ cbc ] a bipartite graph has distinct weakly @xmath5-uniform iasi and strongly @xmath5-uniform iasi if and only if @xmath5 is a positive composite integer . figure [ su - graphs ] illustrates some strong uniform iasis for bipartite graphs . it may be noted that the second graph in figure [ su - graphs ] is a 4-uniform iasi as well as a @xmath88-completely uniform iasi . the following theorem discusses a condition required for a complete graph to admit a strongly @xmath5-uniform iasi . [ tcg ] a strongly @xmath5- uniform iasi of a complete graph @xmath29 is a @xmath89-completely uniform iasi , where @xmath90 . since all distinct pairs of vertices are adjacent in @xmath29 , @xmath29 admits a strongly @xmath5- uniform iasi when the set - indexing number of all vertices of @xmath29 are equal , say @xmath91 . therefore , @xmath29 admits @xmath89-completely uniform iasi . here @xmath92 .    to proceed further , we need the following results from the number theory . [ r8 ] @xcite if @xmath93 a non - square integer , then for every divisor @xmath94 of @xmath93 there exists another divisor @xmath95 for @xmath93 . that is , the number of divisors of a non - square integer is even . [ r9 ] @xcite if @xmath93 a perfect square integer , then for every divisor @xmath94 of @xmath93 there exists another divisor @xmath95 for @xmath93 except for the divisor @xmath96 . that is , the number of divisors of a perfect square is odd .    in view of proposition [ r8 ] and proposition [ r9 ] , we have the following theorem .    [ m3 ] let @xmath1 be a graph which admits a strongly @xmath5-uniform iasi . also , let @xmath60 be the number of divisors of @xmath5 . then if @xmath5 is a non - square integer , then @xmath1 has at most @xmath97 bipartite components and if @xmath5 is a perfect square integer , then @xmath1 has at most of @xmath98 components in which at most @xmath99 components are bipartite components . let @xmath1 be the graph which admits a strongly @xmath5-uniform iasi . also let @xmath100 . * case 1 : * let @xmath5 be a non - square integer . then , by result [ r8 ] , the number of divisors @xmath60 of @xmath5 is even . by remark [ r7 ] , @xmath101 can be partitioned into at most @xmath60 sets . let this partition be @xmath102 , where each @xmath103 is the set of vertices having set - indexing number @xmath104 . since @xmath1 admits a maximal @xmath5-uniform iasi , each vertex of @xmath103 is adjacent to some vertices of @xmath105 so that @xmath106 . also , no two vertices of @xmath103 and no two vertices of @xmath105 are adjacent and no two vertices of both the sets can not be adjacent to the vertices of any other set @xmath107 of @xmath1 . that is , @xmath108 is a component of @xmath1 which is bipartite . we can find at most @xmath97 such bipartite components @xmath109 of @xmath1 . * case 2 : * let @xmath5 be a perfect square . then , by result [ r9 ] , the number of divisors @xmath60 of @xmath5 is odd . as in case 1 , for each @xmath103 of @xmath101 , we can find a set @xmath105 such that @xmath109 is a bipartite component of @xmath1 except for the set @xmath110 whose corresponding divisor @xmath111 . but no vertex of @xmath110 can be adjacent to a vertex of any other set @xmath112 . hence , the induced subgraph @xmath113 $ ] of @xmath1 is a component of @xmath1 such that the vertices in @xmath110 are adjacent to some vertices of itself . therefore , @xmath1 has at most @xmath99 bipartite components and the component @xmath113 $ ] . that is , @xmath1 has at most @xmath98 components of which @xmath99 are bipartite components . [ csb ] let @xmath5 be a non - square integer . then a graph @xmath1 admits a strongly k - uniform iasi if and only if @xmath1 is bipartite or a union of disjoint bipartite components . recall that a _ clique _ of a graph @xmath1 is a maximal complete subgraph of @xmath1 . [ m4 ] if a graph @xmath1 , which admits a strongly @xmath5-uniform iasi , then it contains at most one component which is a clique . let the graph @xmath1 has a strongly @xmath5-uniform iasi . hence , if @xmath1 has a component which is a clique , then @xmath5 is a perfect square .    in view of the results discussed above , we establish the following theorem .    [ tnb ] a connected non - bipartite graph @xmath1 admits a strongly @xmath5-uniform iasi if and only if @xmath5 is a perfect square and this iasi is a @xmath89-completely uniform , where @xmath90 .    invoking the above results , more generally we have    a connected graph @xmath1 admits a strongly @xmath5-uniform iasi if and only if @xmath1 is bipartite or this iasi is a @xmath89-completely uniform iasi of @xmath1 , where @xmath114 . in this paper , we have given some characteristics of the graphs which admit strong iasis and strongly @xmath5-uniform iasis . more properties and characteristics of weak and strong iasis , both uniform and non - uniform , are yet to be investigated . more studies may be done in the field of iasi when the ground set @xmath115 is finite instead of the set @xmath0 . we have formulated some conditions for certain graphs to admit weakly and strongly uniform iasis . the problems of establishing the necessary and sufficient conditions for various graphs and graph classes to admit these types of uniform iasis have still been unsettled . certain iasis which assign set - labels having specific properties , to the elements of given graphs also seem to be promising for further investigations . all these facts highlight the wide scope for intensive studies in this area . 25 b d acharya , * set - valuations and their applications * , mri lecture notes in applied mathematics , the mehta research institute of mathematics and mathematical physics , new delhi,1983 . b d acharya , _ set - indexers of a graph and set - graceful graphs _ , bull . allahabad math . soc . , * 16*(2001 ) , 1 - 23 . b d acharya , k a germina and t m k anandavally , _ some new perspective on arithmetic graphs _ in * labeling of discrete structures and applications * , ( eds . : b d acharya , s arumugam and a rosa ) , narosa publishing house , new delhi , ( 2008 ) , 41 - 46 . b d acharya and k a germina , _ strongly indexable graphs : some new perspectives _ , adv . modeling and optimisation , * 15*(1)(2013 ) , 3 - 22 . s alexander ( 2008 ) , * the mathematical coloring book * , springer - verlag . t m k anandavally , _ a characterisation of 2-uniform iasi graphs _ journal of contemp . sciences , * 8*(10)(2013 ) , 459 - 462 . tom m apostol , * introduction to analytic number theory * , springer - verlag , new york , ( 1989 ) . j a bondy and u s r murty , * graph theory * , springer , ( 2008 ) . j a bondy and u s r murty , * graph theory with applications*,north - holland , new york , ( 1976 ) . d m burton , * elementary number theory * , tata mcgraw - hill inc . , new delhi , ( 2007 ) . g chartrand and p zhang , * introduction to graph theory * , mcgraw - hill inc . , ( 2005 ) . r diestel , ( 2000 ) . * graph theory * , springer - verlag , new york , ( 2000 ) . j a gallian , _ a dynamic survey of graph labelling _ , the electronic journal of combinatorics ( ds 16 ) , ( 2011 ) . k a germina , * set - valuations of a graph and applications * , final technical report , dst grant - in - aid project no.sr/s4/277/05 , the dept . of science and technology ( dst ) , govt . of india , ( 2011 ) . k a germina and kumar abhishek , _ set - valued graphs i _ , ispacs journal of fuzzy set valued analysis , * 2012 * , article idjfsva-00127 , ( 2012 ) , 17 pages . k a germina and t m k anandavally , _ integer additive set - indexers of a graph : sum square graphs _ , journal of combinatorics , information and system sciences , * 37*(2 - 4)(2012 ) , 345 - 358 . k a germina , n k sudev , ( 2013 ) . _ on weakly uniform integer additive set - indexers of graphs _ , international mathematical forum , * 8*(37 ) , 1827 - 1834 . j. gross , j. yellen , * graph theory and its applications * , crc press , ( 1999 ) . f harary , * graph theory * , addison - wesley publishing company inc . , ( 1969 ) . k d joshi , * applied discrete structures * , new age international , ( 2003 ) . m b nathanson ( 1996 ) . * additive number theory , inverse problems and geometry of sumsets * , springer , new york . f s roberts , * graph theory and its practical applications to problems of society * , siam , philadelphia , ( 1978 ) . a rosa , _ on certain valuation of the vertices of a graph _ , in * theory of graphs * , gordon and breach , new york and dunod , paris , ( 1967 ) [ proceedings of the international symposium held in rome ] . d b west , * introduction to graph theory * , pearson education inc . , ( 2001 ) .
let @xmath0 be the set of all non - negative integers . 
 an integer additive set - indexer of a graph @xmath1 is an injective function @xmath2 such that the induced function @xmath3 defined by @xmath4 is also injective . 
 an iasi is said to be _ @xmath5-uniform _ if @xmath6 for all @xmath7 . in this paper 
 , we introduce the notions of strong integer additive set - indexers and initiate a study of the graphs which admit strong integer additive set - indexers . 
 keywords : set - indexers ; integer additive set - indexers ; strong integer additive set - sndexers ; strongly uniform integer additive set - indexers . 
 mathematics subject classification 2010 : 05c78