Topological Invariance under Line Graph Transformations

It is shown that the line graph transformation G L(G) of a graph G preserves an isomorphic copy of G as the nerve of a finite simplicial complex K which is naturally associated with the Krausz decomposition of L(G). As a consequence, the homology of K is isomorphic to that of G. This homology invariance algebraically confirms several well known graph theoretic properties of line graphs and formally establishes the Euler characteristic of G as a line graph transformation invariant.


Introduction
Because of its intrinsic interest, the line graph transformation G L(G) of a graph G has been widely studied.The impetus for much of this research was provided by Ore's discussion of line graphs and problems associated with them [1].Line graphs are also interesting from a practical standpoint, since it has been shown that certain NP-complete problems for graphs are polynomial time problems for line graphs, e.g., [2].Because of their utility for recognizing non-isomorphic graphs, graph invariants have also been the object of intensive research, e.g., [3].
In this paper, a new topological invariance associated with the line graph transformation is found using the natural relationship between a Krausz decomposition of L(G) and an abstract simplicial complex K.In particular, it is shown that, under the line graph transformation, an isomorphic copy of G is preserved as the nerve of K.As a consequence, the homology of G is isomorphic to that of K and an application of the Euler-Poincare formula yields the Euler characteristic of G as a line graph transformation invariant.This invariance also algebraically confirms several well-known graph theoretic properties of line graphs.

OPEN ACCESS
The remainder of this paper is organized as follows: The relevant definitions and terminology are summarized in the next section.Required preliminary lemmas are provided in Section 3 and the main results are established in Section 4. A simple illustrative example is presented in Section 5. Closing remarks comprise the final section of this paper.

Definitions and Terminology
The number of edges incident to a vertex v is the valency of v.A vertex of valency zero is an isolated vertex (only graphs without isolated vertices are considered here).A component of G is a maximally connected subgraph of G.A complete graph K n on n vertices has every pair of vertices adjacent.When if there is an adjacency preserving bijective map Associate with any non-empty graph G its line graph L(G) which has E(G) as its vertex set and has as its edge set those pairs in E(G) which are adjacent in G.A collection of subgraphs of a graph F is a Krausz decomposition of F if (i) each member of is a complete graph; (ii) every edge of F is in exactly one member of ; and (iii) every vertex of F is in exactly two members of .A nonempty graph is a line graph if, and only if, it has a Krausz decomposition and-provided that G 1 and G 2 are non-trivial connected graphs-L(G 1 ) L(G 2 ) if, and only if, G 1 G 2 or {G 1 ,G 2 } is (up to isomorphism) the unordered pair {K 3 ,K 1,3 } [4].
A hypergraph is a pair ( , ), where is a finite set of vertices and is a set of hyperedges which are non-empty subsets of .A Krausz hypergraph of a line graph L(G) has V(L(G)) as its vertex set and the family of subsets of V(L(G)) that induce the members of as its hyperedges.
If S is a finite set, then the closure Cl(S) of S is the family of non-empty subsets of S. The closure Cl( ) of is the union of the closures of its hyperedges, i.e., Cl( ) = E C (E).The number of sets of cardinality k in Cl( ) is h k and is the maximum k for which h k 0.
Let {a 0 ,, a k } be a set of geometrically independent points in n .The k-simplex (or simplex) σ k spanned by {a 0 ,, a k } is the set of points x n for which there exist non-negative real numbers face of σ k is any simplex spanned by a non-empty subset of {a 0 ,, a k }.A finite geometric simplicial complex (or complex) K is a finite union of simplices such that: (i) every face of a simplex of K is in K; and (ii) the non-empty intersection of any two simplices of K is a common face of each.Here it is assumed that all simplicial complexes are finite.Consequently, the dimension of K is the largest positive integer m such that K contains an m-simplex.The vertex scheme of K is the family of all vertex sets which span the simplices of K.The n-skeleton of K is the set of all simplices in K with dimension n.K is connected if, and only if, its 1-skeleton is connected.If {L i } is a family of subcomplexes of K, then i L i and i L i ≠ Ø are subcomplexes of K.
A finite abstract simplicial complex (or abstract complex) is a finite family of finite non-empty sets such that if A is in , then so is every non-empty subset of A. Thus, the vertex scheme of a complex is an abstract complex as are finite unions of set closures and finite intersections of set closures when they are non-empty.
Two abstract complexes and are isomorphic if there is a bijection φ from the vertex set of onto the vertex set of such that {a 0 ,…, a k } if, and only if, {φ(a 0 ),…, φ(a k )} .Every abstract complex is isomorphic to the vertex scheme of some geometric simplicial complex K-in which case K is the geometric realization of and is uniquely determined (up to linear isomorphism).An isomorphism between and the vertex scheme of K is denoted K. To each simplicial complex K there corresponds a chain complex, i.e., abelian groups p (K) and homomorphisms Complexes K and K are homologically isomorphic when H p (K) H p (K ), p 0, and K is homologically acyclic (or acyclic) if H p (K) 0, p 1. The complex of a simplex is acyclic and if K is empty, then K is acyclic.The number of components of K is the betti number b 0 (K).
A cover of a simplicial complex K is a family of subcomplexes = {L α : where A is an index set.The family is an acyclic cover if each L α and each finite intersection α L α are acyclic.The nerve N( ) of is the simplicial complex having A as its vertex set with ∆ = {α 0 ,…,α n } a simplex in N( ) if ∆ L α Ø.

Preliminary Lemmas
The following lemmas are required to prove the main results in the next section.The first four are well known and are stated without proof for completeness.

Lemma 2. [6] A non-empty connected graph G is a tree if, and only if, G is homologically acyclic and H 0 (G)
.

Lemma 3. [7] Let F be a graph. Then F L(G) for some graph G if, and only if, the vertices of G can be placed into one-to-one correspondence with the members of a Krausz decomposition of F such that two vertices of G are adjacent if, and only if, the corresponding members of have a common vertex.
Lemma 4. [8] (Folkman-Leray) If is an acyclic cover of a simplicial complex K, then K and N( ) are homologically isomorphic.
The closure operation C is important for proving the main results of this paper.The required key properties of C are provided by the next lemma.Since the proof is straight forward it is omitted.

Lemma 5. Let {E m : m I } be a collection of non-empty finite sets. Then the following statements are true:
( Lemma 6.Let = ( , ) be a Krausz hypergraph.Suppose = Cl( ) = E C (E) is the abstract complex associated with and its geometric realization is the Krausz complex K.If E i , E j , E k are distinct and the subcomplex K i of K corresponds to the abstract complex Cl(E i ) of , then Proof.Condition (1) follows since E i and E j are induced by a Krausz decomposition and have at most one vertex in common (apply (2) of Lemma 5 with A = E i and B = E j ).Condition (2) follows since no three hyperedges of have a common vertex (apply (3) of Lemma 5 with A = E i E j E k = Ø and then (1) of Lemma 5 with m E m = E i E j E k ).

Main Results
The terminology and results of the previous sections are now used to prove the following main results of this paper.In what follows, it is assumed that: (i) K is a Krausz complex associated with a Krausz hypergraph = ( , ) of a graph

Theorem 1. G N( ).
Proof.By definition of , | | = | |.Also, K i K j ≠ Ø if, and only if, the corresponding pair of hyperedges in have a common vertex.Since F L(G), then Lemma 3 yields a correspondence between V(G) and such that u adjacent to v in G if, and only if, corresponding hyperedges have a vertex in common.Therefore, it follows from the definition of nerve that G N( ).(Recall that here G is assumed to not be isomorphic to K 3 or K 1,3 ).

Lemma 7. is an acyclic cover of K.
Proof.By definition K = i K i , where each K i is the complex of a simplex.Thus, covers K. Since each K i and (via Lemma 6) every finite intersection of the K i 's is acyclic, then is an acyclic cover of K.

Corollary 1.
If G is a (p,q) graph and F L(G), then Proof.The left hand side of the Euler-Poincaré Formula (1), first for K = K and then for K = G, may be equated since, from Theorem 2, the corresponding right hand sides are equal.But when G = K, the left hand side of Equation ( 1) is p − q.Also, because of the one-to-one correspondence between the k − 1 dimensional simplices of K and the sets of size k in = Cl( ) it is the case that the dimension of K is − 1 and k−1 (K) = h k .The validity of Equation ( 2) now follows from these observations and the appropriate K = K and K = G substitutions in the left hand side of Equation ( 1).
Proof.Since L(G) is connected, an application of Corollary 2 shows that G is connected.A connected (p,q) graph G is a tree if, and only if, p − q = 1.The result follows from Equation (2).

Closing Remarks
It has been shown that a Krausz decomposition of the line graph of a graph G defines both an abstract simplicial complex and an acyclic cover of a geometric realization K of the complex such that: (i) the nerve of is isomorphic to G (i.e., the line graph transformation of G preserves an isomorphic copy of G as the nerve of ); and (ii) K and G are homologically isomorphic (i.e., the line graph transformation of G preserves the homology of G as the homology of K).Item (ii) algebraically confirms the graph theoretic fact that G and L(G) have the same number of components when G has no isolated vertices.Thus, it establishes the Euler characteristic of G as a line graph transformation invariant and provides Equation (3) as a condition that must be satisfied by the abstract simplicial complex Cl( ) associated with a Krausz decomposition of a line graph of G when G is a connected tree.It is also interesting to note that Corollary 3 is an algebraic analogue of Rao's Theorem [9].

Figure 2 .
Figure 2. The nerve of the acyclic cover of the Krausz complex associated with L(G).
finite and η p (K) is the number of p-simplices in K, then the rank of p (K) is η p (K) and p (K) is isomorphic to (here denotes both group and graph isomorphism) the direct sum of η p (K) copies of the additive group of integers .The p th homology group of K is the quotient group H p (K) ker ∂ p /im ∂ p 1 and its rank is the p th betti number b p (K).