International Journal of Scientific & Engineering Research, Volume 4, Issue 5, May-2013 2160 ISSN 2229-5518 Ionic Graph and Its Application in Chemistry Azizul Hoque, Nwjwr Basumatary Abstract— A covalent compound can be represented by molecular graph but this graph is inadequate to represent ionic compound. In this paper we introduce some graphs to furnish a representation of ionic compounds. We also study mathematically about these graphs and establish some results based on their relations and structures. Index Terms— Bipartite digraph, Ionic compound, Ionic graph, Graph of an ionic compound, Graph isomorphism, Semi discrete graph, Widen ionic graph. —————————— —————————— 1 INTRODUCTION raph theory is one of the most important area in mathe- compounds. This representation is known as molecular graph matics which is used to study various structural models [1, 2, 4] and is especially true in the case of molecular com- Gor arrangements. This structural arrangements of various pounds known as alkenes. Thus a molecular graph is a repre- objects or technologies lead to new inventions and modifica- sentation of the structural formula of a chemical compound in tions in the existing environment for enhancement in those terms of graph theory. A molecular graph [1, 2, 4] is defined as fields. The field of graph theory started its journey from the a graph whose vertices are corresponds to atoms and edges problem of Konigsberg bridge in 1735 [3]. Graph theoretical are corresponds to chemical bonds between the respective at- ideas are highly utilized by the applications in computer sci- oms. Molecular graphs can distinguish between structural ences [10], especially in research areas of computer sciences isomers, compounds which have the same molecular formula such as data mining, image segmentation, clustering, image but non-isomorphic graphs such as isopentane and neope- capturing, networking etc. For example, a data structure can ntane. On the other hand, the molecular graph normally does be designed in the form of tree. Similarly modelling of net- not contain any information about the three dimensional ar- work topologies can be done using graph theoretical concepts. rangement of the bonds, and therefore cannot distinguish be- In the similar way the most important concept of coloring in tween conformational isomers. Molecular graphs are inade- graph is utilized in resource allocation, scheduling etc. Also, quate to represent ionic compounds. So we have introduced a paths, walks and circuits in graph theory are used in tremen- new family of graphs in such a way that ionic compounds can dous applications such as travelling salesman problem, data- easily be represented. These newly define graphs are dealing base design concepts and resource networking. This leads to with the outer shell of the atoms those are forming ionic com- the development of new algorithIJSERms and theorems that can be pounds. Some of the mathematical and graph theoretical basis used in tremendous applications. In Biology, transcriptional and applications of these graphs are presented in this paper. regulatory networks and metabolic networks would usually be modelled as directed graphs. For instance, in a transcrip- 2 PRELIMINARIES tional regulatory network, nodes represent genes with edges Definition 2.1 A digraph ( , ) is a combination of a non- denoting the interactions between them. As each interaction empty set V and a subset × . The set V is called vertex has a natural associated direction, such networks are modelled 퐷 푉 퐸 set and the set E is called (directed) edge or arc set of D. We as directed graphs. Mason and Verwoerd [9] present a survey write = for an edge with퐸 ⊆ 푉initial푉 point and final point . of the use of graph theoretical techniques in Biology. Especial- The inverse of the edge e is defined as = . In this case, ly, they have used graph theoretical techniques in identifying 푒 푢푢 푢 푢 and are distinct edges. −1 and modelling the structure of bio-molecular networks. Graph 푒 푢푢 Definition−1 2.2 [7] The indegree and the out degree of a vertex theoretical approaches are widely used in Chemistry to study 푒 푒 in a digraph D with vertex set V are defined as the structures and properties of molecular compounds. This ( ) = |{ : = , }| and 푢 idea makes a bridge between chemistry and graph theory and −( ) = |{ : = , }| gave the birth of Chemical graph theory [5]. Chemical graph A푑+lso푢 in [6],푒 푒 ( )푢푢+ 푢 ∈( 푉) = ( ), theory is used to mathematically represent the molecules in D푑 efinition푢 푒2.푒3− A 푢푢digraph푢+ ∈ 푉 is bipartite digraph if the vertex set order to gain insight into the physical properties such as bond- 푑 푢 푑 푢 푑 푢 ∀푢 ∈ 푉 can be partitioned into two non-empty disjoint subsets ing type, boiling point, melting point etc. of these molecular 퐷 and such that any two vertices in the same set are non- 푉퐷 ———————————————— adjacent. 푉1 푉2 • Azizul Hoque is currently pursuing Ph. D. in Mathematics in Gauhati Definition 2.4 An oriented graph is a digraph without loops University, Guwahati, Assam, India-781014. E-mail: [email protected] and multiple edges. • Nwjwr Basumatary is currently pursuing B.S. degree in Mathematical Sciences, IST-GU, Guwahati, Assam, India-781014 Definition 2.5 A graph G on n vertices is called a semi discrete IJSER © 2013 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 4, Issue 5, May-2013 2161 ISSN 2229-5518 graph if either ( ) = 0 or ( ) = 0, for 1 . According to the terminology of this definition, the com- − + Definition 2.6 A퐺 graph푘 G is a 퐺connected푘 graph if there exist a pounds aluminium chloride and scandium fluoride path between any푑 two푢 vertices푑 of푢 G. ≤ 푘 ≤ 푛 are cographic. We think that the class of cographic compounds 퐴퐴퐴퐴3 푆푆퐹3 Definition 2.7 Vertex Chromatic number of a graph G is the has some common properties and characteristics. Thus the minimum number of colors required to colour the vertices of class of cographic compounds must be of fundamental interest G. to chemists and graph theorists. Definition 2.8 An isomorphism of graph G onto a graph H is a Definition 3.4 A subgraph, S of a graph G is said to be ionic bijection : such that any two vertices u and v of G subgraph if there exists an ionic compound C such that are adjacent in G if and only if ƒ(u) and ƒ(v) are adjacent in H. ( ). 푓 푉퐺 ⟶ 푉퐻 푆 ≅ Definition 2.9 The outermost shell of an atom is called the va- Definition 3.5 A graph is said to be widen ionic graph if it has 퐷 퐴 lence shell of the atom. atleast one ionic subgraph. A graph is said to be simple widen Definition 2.10 A cation is a positively charge atom or group of ionic graph if it has no ionic subgraph having more than one atoms obtained by losing some electrons from its valence shell. edge. A graph is said to be strong widen ionic graph if it has An anion is a negatively charge atom or group of atoms ob- atleast one ionic subgraph having more than one edge. One tained by gain some electrons and place them in its valence can easily showed that every digraph is simple widen ionic shell. An ionic compound is a chemical compound in which graph but may not be strong widen ionic graph. ions are held together by the electrostatics forces between cati- ons and anions, and net positive and net negative charges are Theorem 3.1 The graph of an ionic compound is semi discrete equal. For examples, NaCl and AlF3. graph. Lemma 2.1 [Theorem 3.1 of [8]] Semi discrete graphs are bipar- Proof: Let C be an ionic compound and D(C) be the graph of C. tite digraphs. Then Lemma 2.2 [Corollary 3.1 of [8]] Chromatic number of a semi ( ) = { | is an atom in } and discrete graph is two. ( ) = { | , and }. 푉퐷 퐶 푢 푢 퐴 Lemma 2.3 [Theorem 3.2 of [8]] The Square of a semi discrete Thus for any ( ), ( ) = 0 and ( ) = 0 because of 퐸퐷 퐶 푢푢 푢 푢 ∈ 푉 푢 ↝ 푢 graph is the graph itself. the atom can only donate− electrons and +the atom can only 푢푢 ∈ 퐸퐷 퐶 푑 푢 푑 푢 accept electrons. Therefore either ( ) = 0 or ( ) = 0, 푢 − + 푢 ( ). Hence D(C) is semi discrete graph. 3 IONIC GRAPH 푑 푢 푑 푢 ∀푢 ∈ Corollary 3.1 An ionic graph is semi discrete graph. 푉퐷 퐶 Definition 3.1 A graph of an ionic compound C, is a digraph Proof: Let G be an ionic graph. Then there exists an ionic com- ( , ) composed two sets, V the set of all atoms in the com- pound C such that ( ). By theorem 3.1, ( ) is semi pound C and = { | , IJSER and }. By we mean discrete and hence G is also semi discrete. 퐷 푉 퐸 퐺 ≅ 퐷 퐴 퐷 퐴 an electron moves from the valence shell of the atom u to the Corollary 3.2 The graph of an ionic compound is bipartite di- valence shell of퐸 the 푢푢atom푢 푢. ∈The 푉 graph푢 ↝ of 푢 an ionic푢 ↝ compound 푢 C graph. is denoted by ( ). The proof follows from theorem 3.1 and lemma 2.1 푢 Corollary 3.3 An ionic graph is bipartite digraph. 퐷 퐴 Definition 3.2 A graph is said to be ionic graph if it is isomor- The proof follows from corollary 3.1 and lemma 2.1 phic to the graph of an ionic compound. Thus a graph G is Corollary 3.4 The chromatic number of an ionic graph is two. ionic if there exists an ionic compound C such that ( ).