AA Study on the Product Set-Labelingof Graphs
Sudev Naduvath
Centre for Studies in Discrete MathematicsVidya Academy of Science & TechnologyThalakkottukara P. O.,Thrissur - 680501, Kerala, India. [email protected]
Abstract
Let X be a non-empty ground set and P ( X ) be its power set. A set-labeling (or a set-valuation) of a graph G is an injective set-valued function f : V ( G ) → P ( X ) such that the induced function f ∗ : E ( G ) → P ( X ) isdefined by f ∗ ( uv ) = f ( u ) ∗ f ( v ), where f ( u ) ∗ f ( v ) is a binary operation ofthe sets f ( u ) and f ( v ). A graph which admits a set-labeling is known tobe a set-labeled graph. A set-labeling f of a graph G is said to be a set-indexer of G if the associated function f ∗ is also injective. In this paper, weintroduce a new notion namely product set-labeling of graphs as an injectiveset-valued function f : V ( G ) → P ( N ) such that the induced edge-function f ∗ : V ( G ) → P ( N ) is defined as f ∗ ( uv ) = f ( u ) ∗ f ( v ) ∀ uv ∈ E ( G ), where f ( u ) ∗ f ( v ) is the product set of the set-labels f ( u ) and f ( v ), where N is theset of all positive integers and discuss certain properties of the graphs whichadmit this type of set-labeling. Key words : Set-labeling of graphs; product set-labeling of graphs; uniform productset-labeling of graphs; geometric product set-labeling of graphs.
Mathematics Subject Classification : 05C78.
For all terms and definitions, not defined specifically in this paper, we refer to[2, 3, 5, 10]. Unless mentioned otherwise, all graphs considered here are simple,finite, undirected and have no isolated vertices.Let X be a non-empty set and P ( X ) be its power set. A set-labeling (or a set-valuation ) of a graph G is an injective function f : V ( G ) → P ( X ) such that theinduced function f ⊕ : E ( G ) → P ( X ) is defined by f ⊕ ( uv ) = f ( u ) ⊕ f ( v ) ∀ uv ∈ E ( G ), where ⊕ is the symmetric difference of two sets. A graph G which admits aset-labeling is called a set-labeled graph (or a set-valued graph)(see [1]).1 a r X i v : . [ m a t h . G M ] J a n A Study on the Product Set-Labeling of Graphs A set-indexer of a graph G is an injective function f : V ( G ) → P ( X ) such thatthe induced function f ⊕ : E ( G ) → P ( X ) is also injective. A graph G which admitsa set-indexer is called a set-indexed graph (see [1]).Several types of set-valuations of graphs have been introduced in later literatureand the properties and structural characteristics of such set-valued graphs havebeen studied in a rigorous manner. A relevant and important set-labeling in thiscontext is the integer additive set-labeling of graphs which is defined as an injectiveset-valued function f : V ( G ) → P ( X ) such that the induced edge function f + : E ( G ) → P ( X ) is defined by f + ( uv ) = f ( u ) + f ( v ) ∀ uv ∈ E ( G ), where X is a non-empty set of non-negative integers and f ( u ) + f ( v ) is the sumset of the set-labels f ( u ) and f ( v ). Certain types of integer additive set-labeled graphs are studied in[4, 7, 8, 9].Motivated by these studies on different types of set-valuations of graphs, inthis paper, we introduce a new type of set-labeling, namely product set-labeling ofgraphs and study the properties and characteristics of the graphs which admit thistype of set-labeling. Let A and B be two sets of integers. Then, the product set of A and B , denoted by A ∗ B , is the set defined by A ∗ B = { ab : a ∈ A, b ∈ B } . Note that A ∗ ∅ = ∅ and A ∗ { } = { } . Also, if either A or B is a countably infinite set, then their productset is also countably infinite. In view of these facts, we restrict our studies to thenon-empty finite sets of positive integers.Analogous to the corresponding results on sumsets of sets of integers (see [6]),we have the following result on the cardinality of the product set of two sets ofpositive integers. Theorem 2.1. If A and B are two non-empty finite sets of positive integers, then | A | + | B | − ≤ | A ∗ B | ≤ | A | | B | . Theorem 2.2.
For any two sets A and B of positive integers, | A ∗ B | = | A | + | B |− if and only if A and B are geometric progressions with the same common ratio. Using the above mentioned concepts of product sets of sets of positive integers, weintroduce the notion of the product set-labeling of a graph as given below.
Definition 2.1.
Let N be the set of all positive integers and P ( N ) be its powerset. The product set-labeling of a graph G is an injective set-valued function f : V ( G ) → P ( N ) such that the induced edge-function f ∗ : V ( G ) → P ( N ) is definedas f ∗ ( uv ) = f ( u ) ∗ f ( v ) ∀ uv ∈ E ( G ), where f ( u ) ∗ f ( v ) is the product set of theset-labels f ( u ) and f ( v ). A graph G which admits a product set-labeling is calleda product set-labeled graph . Definition 2.2. A product set-labeling f : V ( G ) → P ( N ) of a graph G is said tobe a product set-indexer if the induced edge-function f ∗ : V ( G ) → P ( N ) defined by f ∗ ( uv ) = f ( u ) ∗ f ( v ) ∀ uv ∈ E ( G ) is also an injective function. udev Naduvath G iscalled label size of that element. A product set-labeling f of a graph G is said tobe a uniform product set-labeling if all edges of G have the same label size under f . In particular, a product set-labeling f of a graph G is said to be k -uniform if | f ∗ ( uv ) | = k ∀ uv ∈ E ( G ).In view of Theorem 2.1, the bounds for the label size of edges of a productset-labeled graph G is given by | f ( u ) | + | f ( v ) | − ≤ | f ∗ ( uv ) | = | f ( u ) ∗ f ( v ) | ≤ | f ( u ) | | f ( v ) | ∀ uv ∈ E ( G ) . (1)The product set-labelings which satisfy the bounds of this inequality are ofspecial interest. If the cardinality of the vertex set-labels of G under a product set-labeling f attains the upper bound of the inequality (1), then f is called a strongproduct set-labeling of G . Before proceeding to investigate the conditions for theexistence of a strong product set-labeling, we require the following notion. Definition 2.3.
Let A be a non-empty set of positive integers. Then the quotientset of the set A , denoted by Q A , is defined as Q A = { ab : a, b ∈ A, a ≥ b } . That is, Q A is the set of all rational numbers, greater than or equal to 1, which is formedby the elements of the set A .In view of this notion, we establish a necessary and sufficient condition for agraph G to admit a strong product set-labeling in the following theorem. Theorem 2.3.
A product set-labeling f of a graph G is a strong product set-labelingif and only if the quotient sets of the set-labels of any pair of adjacent vertices of G are disjoint.Proof. Let f be a product set-labeling of a graph G and let u and v be any twoadjacent vertices in G .First, assume that f is a strong product set-labeling of G . Then, we have | f ( u ) ∗ f ( v ) | = | f ( u ) | | f ( v ) |∀ uv ∈ E ( G ). This is possible only when ac (cid:54) = bd for anytwo distinct elements a, b ∈ f ( u ) and any two distinct elements c, d ∈ f ( v ). Thatis, ab (cid:54) = cd . Since ab ∈ Q f ( u ) and cd ∈ Q f ( v ) , we have Q f ( u ) ∩ Q f ( v ) = ∅ .If possible let f is not a strong product set-labeling of G . Then | f ( u ) ∗ f ( v ) | < | f ( u ) | | f ( v ) | for some uv ∈ E ( G ). That is, there exist at least two elements a, b ∈ f ( u ) and at least two elements c, d ∈ f ( v ) such that ac = bd . That is, ab = cd .Hence, Q f ( u ) ∩ Q f ( v ) (cid:54) = ∅ . This completes the proof.By saying that a set is a geometric progression, we mean that the elements ofthat set is in geometric progression. If the context is clear, the common ratio of theset-label of an element (a vertex or an edge) in G may be called the common ratioof that element .In view of Theorem 2.2, we also note that the vertex set-labels in G , underthe product set-labeling f which attains the lower bound the inequality (1), aregeometric progressions having the same common ratio. This fact creates muchinterest in investigating the set-labels of the elements of G which are geometricprogressions. Hence we have the following notion. A Study on the Product Set-Labeling of Graphs
Definition 2.4.
A product set-labeling f : V ( G ) → P ( N ) of a graph G is said to bea geometric product set-labeling if the set-labels of all elements (vertices and edges)of G with respect to f are geometric progressions.The following theorem discusses the conditions required for a product set-labeling f of a graph G to be a geometric product set-labeling of G . Theorem 2.4.
A product set-labeling f : V ( G ) → P ( N ) of a graph G is a geometricproduct set-labeling of G if and only if for every edge of G , the common ratio ofone end vertex is a positive integral power of the common ratio of the other endvertex, where this power is less than or equal to the label size of the end vertexhaving smaller common ratio.Proof. Let f be a product set-labeling of a graph G under which every vertex set-label is a geometric progression and let u and v be any two adjacent vertices of G .Let r u and r v be the common ratios of u and v respectively such that r u ≤ r v . Let theset-labels of u and v be given by f ( u ) = { a i = a ( r u ) i − : a ∈ N ; 0 ≤ i ≤ | f ( u ) | = m } and f ( u ) = { b j = b ( r v ) j − : b ∈ N ; 0 ≤ j ≤ | f ( v ) | = n } . Now, consider the followingsets. A = f ( u ) ∗ { b } = { ab, abr u , abr u , . . . , ab ( r u ) m − } A = f ( u ) ∗ { b } = { abr v , abr u r v , abr u r v , . . . , ab ( r u ) m − r v } ... ... ... A j = f ( u ) ∗ { b j } = { abr jv , abr u r jv , abr u r jv , . . . , ab ( r u ) m − r jv } ... ... ... A n − = f ( u ) ∗ { b n − } = { abr n − v , abr u r n − v , abr u r n − v , . . . , ab ( r u ) m − r n − v } Here we can see that f ∗ ( uv ) = n − (cid:83) j =0 A j .Now assume that r v = ( r u ) k , for some positive integer k ≤ | f ( u ) | = m . Then,either some of the initial elements of the set A j +1 coincides with some final elementsof A j or the ratio between the first element of A j +1 and the final element of A j is r u , for 0 ≤ j ≤ n −
1. In both cases A j ∪ A j +1 is a geometric progression for all0 ≤ j ≤ n −
1. Hence f ∗ ( uv ) is a geometric progression for all edge uv ∈ E ( G ) andhence f is a geometric product set-labeling of G .If r v = ( r u ) k and k ≥ | f ( u ) | , then for 0 ≤ j ≤ n −
1, we note the following facts(i) A j and A j +1 are geometric progressions with the same common difference r u ,(ii) A j ∩ A j +1 = ∅ ,(iii) A j ∪ A j +1 is not a geometric progression, as the ratio between the first elementof A j +1 and the final element of A j is not equal to r u .Therefore, in this case, f ∗ ( uv ) is not a geometric progression and hence f is not ageometric product set-labeling of G .Now consider the case that r v (cid:54) = ( r u ) k for any positive integer k . Then, A j and A j +1 are geometric progressions with different common ratios and hence it is clear udev Naduvath A j ∪ A j +1 is not a geometric progression. Therefore, in this case also, f ∗ ( uv )is not a geometric progression and hence f is not a geometric product set-labelingof G . This compltes the proof.The following result describes a necessary and sufficient condition for a completegraph to admit a geometric product set-labeling. Corollary 2.5.
A complete graph K n admits a geometric product set-labeling if andonly if the common ratio of every vertex is either an integral power or a root of thecommon ratios of all other vertices of K n .Proof. Since any two vertices in K n are adjacent to each other, the proof follows asan immediate consequence of Theorem 2.4.The characteristic index of an edge e = uv of a product set-labeled graph G isthe number k ≥
1, such that r v = ( r u ) k , where r u and r v are the common ratios ofthe set-labels of the vertices u and v (or equivalently, the common ratios of u and v ) respectively.The following result discusses the label size of the edges of a graph G whichadmits a geometric product set-labeling. Proposition 2.6.
Let f be a geometric product set-labeling of a graph G and let u and v be two adjacent vertices in G with the common ratios r u and r v such that r u ≤ r v . Then, the label size of the edge uv is given by | f ∗ ( uv ) | = | f ( u ) | + k ( | f ( v ) | − ,where k is the characteristic index of the edge uv .Proof. Since f is a geometric product set-labeling of G , by Theorem 2.4, for anyadjacent vertices u and v with the common ratios r u and r v such that r u ≤ r v ,we have r v = ( r u ) k , where k is a positive integer less than or equal to | f ( u ) | . Let f ( u ) = { ar i − u : a ∈ N ; 0 ≤ i ≤ | f ( u ) |} and f ( v ) = { br j − v : b ∈ N ; 0 ≤ j ≤ | f ( v ) |} .Then, we have f ∗ ( uv ) = { abr i − u r j − v : a, b ∈ N ; 0 ≤ i ≤ m, ≤ j ≤ n } = { abr i − u ( r u ) kj − : a, b ∈ N ; 0 ≤ i ≤ m, ≤ j ≤ n } = { abr ( i − k ( j − u : a, b ∈ N ; 0 ≤ i ≤ m, ≤ j ≤ n } . Therefore, | f ∗ ( uv ) | = m + k ( n −
1) = | f ( u ) | + k ( | f ( v ) | − G to be a strong product set-labeling of G . Theorem 2.7.
A geometric set-labeling f of a graph G is a strong product set-labeling of G if and only if the characteristic index of every edge of G is equal tothe label size of its end vertex having smaller common ratio.Proof. Let f be a geometric product set-labeling of G and let u and v be twoadjacent vertices of G with common ratios r u and r v respectively such that r u ≤ r v . A Study on the Product Set-Labeling of Graphs
First, assume that f is also a strong product set-labeling of G . Then, we have | f ∗ ( uv ) | = | f ( u ) | | f ( v ) | ∀ uv ∈ E ( G ). But, by Proposition 2.6, we have | f ∗ ( uv ) | = | f ( u ) | + k ( | f ( v ) | − | f ∗ ( uv ) | = | f ( u ) | + k ( | f ( v ) | − i.e, | f ( u ) | | f ( v ) | = | f ( u ) | + k ( | f ( v ) | − ∴ k = | f ( u ) | | f ( v ) | − | f ( u ) | ( | f ( v ) | − | f ( u ) | . Conversely assume that the characteristic index of every edge of G is equal tothe label size of its end vertex having smaller common ratio. That is, let k = | f ( u ) | .Then, we have | f ∗ ( uv ) | = | f ( u ) | + k ( | f ( v ) | −
1) for all uv ∈ E ( G )= | f ( u ) | + | f ( u ) | ( | f ( v ) | − | f ( u ) | | f ( v ) | Therefore, f is a strong product set-labeling of G . This completes the proof.When the set-labels of two adjacent vertices are geometric progressions with thesame common ratio, then the characteristic index of the edge between them is 1.Invoking this fact, we define a particular type of geometric product set-labeling asfollows. Definition 2.5. An isogeometric product set-labeling of a graph G is a product set-labeling of G with respect to which the set-labels of all elements of G are geometricprogressions with the same common ratio.In view of Theorem 2.2, we note that for any graph G which admits an isoge-ometric product set-labeling, the label size of every edge is one less than the sumof the label sizes of its end vertices and also note that the characteristic index ofevery edge of G is 1.The following is an obvious result on the admissibility of an isogeometric productset-labeling by any given graph. Theorem 2.8.
Every graph admits an isogeometric product set-labeling.Proof.
Let V = { v , v , v , . . . , v n } be the vertex set of the given graph G . Choosetwo sets A = { a i ∈ N : 1 ≤ i ≤ | V | = n } and B = { m i ∈ N : 1 ≤ i ≤ n } . Nowlabel vertices of G by the geometric progression f ( v i ) = { a i , a i r, a i r , . . . , a i r m i − } ;1 ≤ i ≤ n , where r is a positive integer greater than 1. Then, the set-label of anyedge v i v j in G is given by f ∗ ( v i v j ) = { a i a j , a i a j r, a i a j r , . . . , a i a j r m i + m j − } , whichis also a geometric progression with the common ratio r . That is, the set-label ofall elements of G are geometric progressions with the same common ratio r . Hence, f is an isogeometric product set-labeling of G .In the following theorem, we discuss the condition required for an isogeometricproduct set-labeling of a graph G to be a uniform product set-labeling of G . udev Naduvath Theorem 2.9.
An isogeometric product set-labeling of a connected graph G is auniform product set-labeling if and only if any one of the following conditions holds.(i) the label size of all vertices of G are equal.(ii) G is bipartite with label size of vertices in the same partition are equal.Proof. Let f be an isogeometric product set-labeling of a given graph G . If | f ( v ) | = m , a positive integer m for all v ∈ V ( G ), then all edges of G has the label size 2 m − V ( G ) such that | f ( v ) | (cid:54) = m , then assume that G isa bipartite graph with bipartition ( X, Y ) such that all vertices in X have the labelsize m and all vertices in Y have the label size n . Here, by Proposition 2.6, alledges of G have the label size m + n −
1. In both cases, f is a uniform productset-labeling of G .Conversely, assume that the isogeometric product set-labeling f is also a uniformproduct set-labeling of G . If the label size of all vertices of G are equal, then theproof is complete. Hence assume otherwise. Let u be an arbitrary vertex of G which has the label size m . Since f is a uniform geometric product set-labeling, allvertices v in the neighbouring set N ( u ) of the vertex u must have the same labelsize, say n . Using the same argument, all vertices in the neighbouring set of N ( u )must have the label size m . Since G is a connected graph, the vertex set V ( G ) canbe partitioned in to two sets such that the vertices in the first partition have thelabel size m the vertices in the other partition have the label size n . Since m (cid:54) = n ,no two vertices in the same partition are adjacent also. Hence, G is a bipartitegraph with the vertices in the same partition have same label size.We have already noticed that the characteristic index of all edges of a graphwhich admits an isogeometric product set-labeling is 1. But, In general, the char-acteristic indices of all edges of a geometric product set-labeled graph need not bethe same. This fact creates a lot of interest in studying the structural properties ofa geometric product set-labeled graph, all whose edges have the same characteristicindex greater than 1. Hence we have the following notion. Definition 2.6.
A geometric product set-labeling of a graph G is said to be a like-geometric product set-labeling if all edges have the same characteristic index k > G to admit a like-geometric product set-labeling. Theorem 2.10.
A graph G admits a like-geometric product set-labeling if and onlyif it is bipartite.Proof. First, assume that G is a bipartite graph with a bipartition ( X, Y ). Let X = { v i : 1 ≤ i ≤ m } and Y = { u j : 1 ≤ j ≤ n } , where m, n ∈ N . Choose thesets M = { m i ∈ N : m i ≥ , ≤ i ≤ | X | = m } , M = { a i ∈ N : 1 ≤ i ≤ m } , N = { n j ∈ N : n j ≥ , ≤ j ≤ | Y | = n } and N = { b j ∈ N : 1 ≤ j ≤ n } .Let k = min { m i : 1 ≤ i ≤ m } and choose two positive integers r and s such that s = r k . Now, define a product set-labeling f on G which assigns each vertex v i of X A Study on the Product Set-Labeling of Graphs to a geometric progression f ( v i ) = { a i , a i r, a i r , . . . , a i r m i − } ; 1 ≤ i ≤ | X | and eachvertex u j of Y to a geometric progression f ( u j ) = { b j , b j s, b j s , . . . , b j s n j − } ; 1 ≤ j ≤| Y | . Then, by Theorem 2.4, for every edge v i u j in G , if exists, the set-label f ∗ ( v i u j )is a geometric progression with common ratio r and the characteristic index of everyedge in G will be k . Hence, the function f is a like-geometric product set-labelingof G .Next, assume that G is not a bipartite graph. Then, G contains at least one oddcycle. Let C n be such an odd cycle in G . Now, choose two positive integers r and s such that s = r k . Label the vertices of C n with odd subscripts by distinct geometricprogressions with common ratio r and label the vertices with even subscripts bydistinct geometric progressions with common ratio s . Then, all edges except v n v attain the characteristic index k and the edge v n v has the characteristic index 1.In all other labeling of the vertices of G with geometric progressions such that themaximum number of edges attains the characteristic index k , we can see that atleast one edge of C n has the characteristic index k q , for some positive integer q (cid:54) = 1.In all these cases, it is to be noted that f is not a like-geometric product set-labelingof G . This completes the proof.The following proposition provides the condition required for a like-geometricproduct set-labeling of a graph G to be a uniform product set-labeling of G . Theorem 2.11.
A like-geometric product set-labeling of a (bipartite) graph G is auniform product set-labeling of G if and only if the vertices in the same partition of G have the same label size.Proof. Let f be a like-geometric product set-labeling of a connected graph G . Then,by Theorem 2.10, G is bipartite. Let ( X, Y ) be a bipartition of G .First, let all vertices in X have the same label size, say m and all vertices in Y have the same label size, say n . Then, by Proposition 2.6, the label size of all edgesof G is m + k ( n − k ≤ n . Hence, f is a uniform product set-labeling of G .Now, assume that f is also a uniform product set-labeling of G . Then, exactlyas explained in the converse part of the proof of Theorem 2.9, we can partition thevertex set of G in two subsets X and Y such that all vertices in X have the samelabel size, say m and all vertices in Y have the same set-label, say n and such thatno two vertices in the same partition are adjacent to each other. This completesthe proof.A necessary and sufficient condition for a like-geometric product set-labeling ofa graph G to be a strong product set-labeling of G is provided in the followingresult. Theorem 2.12.
A like-geometric product set-labeling f of a graph G is a strongproduct set-labeling of G if and only if all vertices in one partition have the samelabel size.Proof. Let f be a like-geometric product set-labeling of G . Clearly, by Theorem2.10, G is bipartite. Let ( X, Y ) be a bipartition of G . Without loss of generality,label all vertices in X by distinct geometric progressions of same cardinality, say udev Naduvath m and the same common ratio r , where r is any positive integer greater than 1.Now label the vertices in Y by distinct geometric progressions with common ratio r m . Then, by Theorem 2.7, | f ∗ ( uv ) | = | f ( u ) | | f ( v ) | ∀ uv ∈ E ( G ). Therefore, f is astrong product set-labeling of G .Conversely, assume that f is a strong product set-labeling of G . Then, thecharacteristic index k of every edge of G is equal to the cardinality of the set-labelof its end vertex having smaller common ratio. Since f is a like-geometric productset-labeling, the characteristic index of every edge of G is the same and is equal tothe minimum label size of the vertices having smaller common ratio. Hence, thelabel size of all vertices in the corresponding partition are the same. This completesgraph.In view of the above two theorems, we have the following result. Corollary 2.13.
A like-geometric product set-labeling f of a graph G is a stronglyuniform product set-labeling of G if and only if all vertices in the same partitionhave the same label size.Proof. The proof immediately follows from Theorem 2.11 and Theorem 2.12, bytaking the value k = m in the respective proofs. In this paper, we have discussed the characteristics and properties of the graphswhich admit different types of product set-labeling. There are several open problemsin this area. Some of the open problems that seem to be promising for furtherinvestigations are following.
Problem 1.
Characterise the product set-labeled graphs whose vertex set-labelsare geometric progressions but the edge set-labels are not.
Problem 2.
Characterise the product set-labeled graphs whose edge set-labels aregeometric progressions but the vertex set-labels are not.
Problem 3.
Discuss the conditions required for an arbitrary geometric productset-labeling of a graph to be a uniform product set-labeling of G . Problem 4.
Discuss the admissibility of different types of product set-labelings bydifferent graph operations, graph products and graph powers.
Problem 5.
Characterise the product set-labeled graphs in which the label size ofits edges are equal to the label sizes of one or both of their end vertices.Further studies on other characteristics of product set-labeled graphs corre-sponding to different types of product set-labelings are also interesting and chal-lenging. All these facts highlight the scope for further studies in this area.0
A Study on the Product Set-Labeling of Graphs
Acknowledgement
The author would like to dedicate this work to Prof. (Dr.) T. Thrivikraman, whohas been his mentor, motivator and the role model in teaching as well as in research.
References [1] B. D. Acharya,
Set-valuations and their applications , MRI Lecture notesin Applied Mathematics, No.2, The Mehta Research Institute of Mathematicsand Mathematical Physics, Allahabad, 1983.[2] J. A. Bondy and U. S. R. Murty,
Graph theory with application , North-Holland, New York, 1982.[3] J. A. Gallian,
A dynamic survey of graph labelling , Electron. J. Combin.,(2015), (
On weakly uniform integer additive set-indexers of graphs , Int. Math. Forum, (37)(2013), 1827 1834., DOI:10.12988/imf.2013.310188.[5] F. Harary, Graph theory , New Age International, New Delhi., 2002.[6] M. B. Nathanson,
Additive number theory, inverse problems and ge-ometry of sumsets , Springer, New York, 1996.[7] N. K. Sudev and K. A. Germina,