A Note on the Application of Incidence Matrices of Simplicial Complexes

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1 Int. J. Contep. Math. Sciences, Vol. 8, 2013, no. 19, HIKARI Ltd, A Note on the Application of Incidence Matrices of Siplicial Coplexes Veerendra Vikra Awasthi Departent of Matheatics and Statistics Indian Institute of Science Education and Research Kolkata Mohanpur, West Bengal , India vvawasthi@iiserkol.ac.in, vvawasthi@gail.co Copyright c 2013 Veerendra Vikra Awasthi. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. Abstract In conteporary atheatics, the siplicial coplexes are studied in graph theory by considering the as higher diensional graphs. The ain ai of this article is to apply the notion of incidence atrices of oriented siplicial coplexes to copute cobinatorial laplacian of siplicial coplexes directly. Matheatics Subject Classification: 05B20, 05C50, 05E45, 35J05 Keywords: Incidence atrix, Siplicial coplex, Cobinatorial Laplacian 1 Introduction In coputation of cobinatorial laplacian of a siplicial coplex one has to copute the atrix representation of the boundary operator between the chain coplexes of higher orders. As already defined in [3] that the cobinatorial laplacian operator of higher orders say q of an oriented siplicial coplex with diension greater than q 1 is obtained by knowing the atrix representation of siplicial boundary operators q and q 1. In this article we prove that the cobinatorial laplacian of an oriented siplicil coplex can be coputed directly just by coputing the incidence atrices of that order. Thus this

2 936 Veerendra Vikra Awasthi ethod gives a sipler and direct ethod for coputation of the cobinatorial laplacian of oriented siplicial coplex. 2 Definitions For definition of oriented siplicial coplex, oreinted siplexes readers are suggested to refer [1]. Definition 2.1 [1] The incidence nuber associated with two siplexes which differ by 1 in their diension (σ p+1,σ p ) of an oriented siplicial coplex K denoted by [σ p+1,σ p ] is defiend as follows: (a) If σ p is not a face of σ p+1, we put [σ p+1,σ p ]=0. (b) If σ p is a face of σ p+1, This eans all the vertices of σ p are also the vertices of σ p+1 except one vertex as σ p+1 has p + 2 vertices and σ p has p + 1 vertices. Let σ p =<v 0,v 1,,v p > be positively oriented. Let v be the additional vertex of σ p+1. Then putting v as the first vertex <v,v 0,,v p >, we get either σ p+1 is positively oriented or negatively oriented. We define { 1 if σ [σ p+1,σ p ]= p+1 is positively oriented. 1 if σ p+1 is negatively oriented. The general definition for the incidence atrix of a directed graph is given as follows: Definition 2.2 The incidence atrix of a directed graph G is a p q atrix (a ij ) where p and q are the nuber of vertices and edges respectively, such that a ij = 1 if the edge e j leaves vertex v i, (a ij ) = 1 if the edge e j enters the vertex v i and 0 otherwise. Considering a directed graph as 1-diensional oriented siplicial coplex and applying the notion of ncidence nuber associated with vertices and edges of the graph We have an alternative definition for the incidence atrix of graph: Definition 2.3 The incidence atrix of a directed graph G is the atrix I G =(a ij ) n where a ij is the incidence nuber [σj 1,σi 0 ]. Where σj 1 is the directed edge i.e., the 1-diensional siplex and σi 0 is the vertex i.e., the 0-siplex. We generalise the above definition and define the higher order incidence atrices for an oriented siplicial coplexes as follows.

3 Application of incidence atrices of a siplicial coplexes 937 Definition 2.4 Let K be a oriented siplicial coplex of diension k. Let the total nuber of k 1 siplexes be and the total nuber of k siplexes be n Then incidence atrix of order k, I k is defined as follows: ) I k = M n = (a ij Where a ij is the incidence nuber of the pair (σj k,σk 1 i ) i.e., a ij =[σj k,σk 1 for all i =1,, and j =1,,n. 3 Main Result In this section we prove that for a siplicial coplex K with di K q, the atrix representation of a siplicial boundary operator q : C q (K) C q (K 1) can be coputed in ters of the incidence atrix of the siplicial coplex K of order q. Let B q be the atrix representation of the boundary operator q : C q (K) C q 1 (K) relative to the standard bases for C q (K) and C q 1 (K). Let q : C q 1 (K) C q (K) be the adjoint operator associated with the boundary operator q. (See [3] for the definitions of adjoint operator of a boundary operator). Fro ([2], Theore 6.10) the transpose of the atrix for the q i.e., Bq T is the atrix representation of the q the qth adjoint boundary operator with respect to this sae ordered basis. Theore 3.1 [3] Let K be an oriented siplicial coplex with di K q + 1. Then the cobinatorial Laplacian of order q is the linear operator L q (K) :C q (K) C q (K) defined as follows: n i ] L q (K) = q+1 q+1 + q q We prove here that the cobinatorial Laplacian of order q, L q (K) :C q (K) C q (K) can be deterined directly by calculating the incidence atrices of higher orders. Theore 3.2 Let K be an oriented siplicial coplex with di K q. The cobinatorial Laplacian of order q, L q (K) :C q (K) C q (K) is given by the following expression L q (K) =I T q+1 I q+1 + I q I T q where I q+1 and I q are the incidence atrices of order q + 1 and q. Proof : The q-diensional chain coplex C q (K) is a vector space of oriented q-chains on K with coefficients in R. All the q-siplexes of K fors a basis for

4 938 Veerendra Vikra Awasthi C q (K). The boundary operator q : C q (K) C q 1 (K) is defined by extending linearly the following ap on basis eleents. q q (<v 0,v 1,,v q >)= ( 1) i <v 0,v 1,, ˆv i,,v q > (1) i=0 with the property that q 1 q = 0 where ˆv i eans the vertex v i has been oitted. The boundary operator q : C q (K) C q 1 (K) can also be defined as the following [1]. q q (<v 0,v 1,,v q >)= [σ q,σ q 1 i ] <v 0,v 1,, ˆv i,,v q > (2) i=0 where [σ q,σ q 1 i ] is the incidence nuber of the siplices σ q =<v 0,v 1,,v q > and σ q 1 i =< v 0,v 1,, ˆv i,,v q > and σ q 1 i runs over all positively oriented (q 1)-faces of σ q. If the oreinted q siplex σ q =< v 0,v 1,,v q > has ordering v 0 <v 1 < <v q, then it is clear that σ q has (q + 1)-faces naely, <v 0,v 1,, ˆv i,,v q >, for all i =0, 1,...,q, and we observe that each of these faces gets orientation by the orientation of σ q. Thus calculationg the incidence nuber [σ q,σ q 1 i ] we find that the incidence nuber is ( 1) i and, therefore, both the definitions in equations (1) and (2) are equivalent. Let the cardinality of the set of all q-siplices be n i.e., there are {σ1, q,σn q } q-siplexes and the cardinality of the set of all (q 1)-siplices be i.e. {σ q 1 1,,σ q 1 } are all the (q 1)-siplexes of K. The boundary operator q : C q (K) C q 1 (K) defined on the bases eleents of C q (K) in (2) is following q (σ q i )= [σ q i,σq 1 j ]σ q 1 j for all i =1,,n i.e., j=1 q (σ1) q = [σ1,σ q q 1 1 ]σ q [σ1,σ q q 1 q (σ q 2 ) = [σq 2,σq 1 1 ]σ q [σ q 2,σq 1 q (σn q ) = [σq n,σq 1 1 ]σ q [σn q,σq 1 ]σ q 1 ]σq 1 ]σq 1 Thus the atrix B q associated with the boundary operator q is the following [σ1,σ q q 1 1 ] [σ1,σ q q 1 2 ] [σ1,σ q q 1 ] [σ2,σ q q 1 1 ] [σ2,σ q q 1 2 ] [σ2,σ q q 1 ] B q =.. [σn q,σq 1 1 ] [σn q,σq 1 2 ] [σn q,σq 1 ]

5 Application of incidence atrices of a siplicial coplexes 939 Also observe that by the definition of incidence atrix, the incidence atrix of order q, I q is given by [σ q 1,σq 1 1 ] [σ q 2,σq 1 1 ] [σ q n,σq 1 1 ] [σ q 1,σq 1 2 ] [σ q 2,σq 1 2 ] [σn q,σq 1 2 ] I q =.. [σ1,σ q q 1 ] [σ2,σ q q 1 ] [σn,σ q q 1 ] Therefore, the transpose of then incidence atrix of order q, I q is the atrix B q. i.e., = B q I T q The cobinatorial Laplacian of order q, L q (K) :C q (K) C q (K) can be given in the for of the atrix representations of the boundary operators [3]. Let B q+1 and B q be the atrix representations of the boundary operators q+1 and q. Then L q (K) =B q+1 B T q+1 + B qb q. Hence the cobinatorial Laplacian of order q, L q (K) :C q (K) C q (K) is L q (K) =I T q+1 I q+1 + I q I T q. References [1] Deo, Satya, Algebraic topology. A prier, Hindustan Book Agency, India, [2] Friedberg, Stephen H., and Insel, Arnold J., and Spence, Lawrence E., Linear Algebra, third ed., Prentice-Hall, Englewood Cliffs, NJ, [3] Goldberg, Tiothy E., Cobinatorial Laplacians of Siplicial Coplexes, Senior Thesis, Bard College, Received: Septeber 11, 2013