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Discrete Mathematics 308 (2008) 555–564 www.elsevier.com/locate/disc
A generalized characteristic polynomial of a graph having a semifree actionଁ Hye Kyung Kima, Jaeun Leeb
aDepartment of Mathematics, Catholic University of Daegu, Kyungsan 712-702, Republic of Korea bDepartment of Mathematics, Yeungnam University, Kyongsan 712-749, Republic of Korea
Received 20 June 2006; received in revised form 5 March 2007; accepted 15 March 2007 Available online 24 March 2007
Abstract For an abelian group , a formula to compute the characteristic polynomial of a -graph has been obtained by Lee and Kim [Characteristic polynomials of graphs having a semi-free action, Linear algebra Appl. 307 (2005) 35–46]. As a continuation of this work, we give a computational formula for generalized characteristic polynomial of a -graph when is a finite group. Moreover, after showing that the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, we compute the reciprocals of the Bartholdi zeta functions of wheels and complete bipartite graphs as an application of our formula. © 2007 Elsevier B.V. All rights reserved.
MSC: 05C50; 05C25; 15A15; 15A18 Keywords: Characteristic polynomial; Semifree action; Bartholdi zeta function
1. Introduction
Let G be a connected graph with vertex set V (G) and edge set E(G). Let G and εG denote the number of vertices and edges of G. Let D be a connected digraph with vertex set V(D)and edge set E(D). Let D and εD denote the number of vertices and edges of D. The adjacency matrix A(D) = (aij ) of D is the D × D matrix whose (i, j)-entry aij is the number of directed edges from vi to vj and the adjacency matrix A(G) of G is the G × G matrix A(G), where G is the digraph obtained from G by replacing each edge of G with a pair of oppositely directed edges. The degree matrix D(G) of G is the diagonal matrix whose (i, i)-th entry is the degree dG(vi) of vi for each 1i G. The characteristic polynomial of G, denoted by G(), is the characteristic polynomial of A(G).In[2], Cvetkovic et al. introduced a polynomial on two variables of G, FG(, ) = det(I − (A(G) − D(G))) as a generalization of some characteristic polynomials of G. For example, the characteristic polynomial of G is FG(, 0) and the characteristic polynomial of the Laplacian matrix D(G) − A(G) of G is (−1) G FG(−, 1).In[1], a comprehensive overview and problems on the Bartholdi zeta function of a graph were provided. In particular, the reciprocal of the Bartholdi zeta
ଁ Supported by Com2MaC-KOSEF (R11-1999-054). E-mail addresses: [email protected] (H.K. Kim), [email protected] (J. Lee).
0012-365X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2007.03.029 556 H.K. Kim, J. Lee / Discrete Mathematics 308 (2008) 555–564 function of G is given as
−1 2 2 εG−G 2 ZG(u, t) = (1 − (1 − u) t ) det[I − A(G)t + (1 − u)(DG − (1 − u)I)t ].
In Theorem 10, we will show that the polynomial FG(, ) determines the reciprocal of the Bartholdi zeta function of a graph G and vice versa. By |X|, we denote the cardinality of a finite set X. Let be a finite group. We say that G admits a -action if there is a group homomorphism from to Aut(G). For each v ∈ V (G), let v ={ ∈ |(v) = v} be the isotropy subgroup of v, and Fix ={v ∈ V (G)|v = }. We call Fix the fixed part of V (G). We say that acts semifreely on G if for each v ∈ V (G), v is either the trivial group or the full group , and for each e ∈ E(
2. Adjacency matrix of a -graph
In this section, we study the adjacency matrix and degree matrix of a -graph for any group . A map between two digraphs D and D is direction preserving if the initial vertex of every directed edge in D is mapped onto the initial vertex of the image edge in D. A digraph D˜ is called a covering graph of D if there exists a direction f : D˜ → D f : V(D)˜ → V(D) f : E(D)˜ → E(D) preserving map with the following properties: |V(D)˜ and |E(D)˜ are surjective and for each v˜ ∈ V(D)˜ , f maps the set of edges originating (resp. terminating) at v˜ one-to-one onto the set of edges originating (resp. terminating) at f(v)˜ . We call f : D˜ → D a covering and D the base graph.A covering f : D˜ → D is regular if there exists a group of graph automorphisms of D˜ acting freely on D˜ and a graph isomorphism h : D/˜ → D such that the diagram
commutes, i.e., h ◦ q = f , where q is the quotient map. The set of directed edges of G is denoted by E(G) .Bye−1, we mean the reverse edge to an edge e ∈ E(G) .We denote the directed edge e of G by uv if the initial and terminal vertices of e are u and v, respectively. For a finite group ,a-voltage assignment of G is a function : E(G) → such that (e−1) = (e)−1 for all e ∈ E(G) . We denote the set of all -voltage assignments of G by C1(G; ). For general terms, we refer to [4]. In [7], a construction method of -digraphs was introduced as follows: let D be a digraph. For a subset S of V(D), is the subgraph of D induced by S. For a pair of subsets S1 and S2 of V(D), E(S1,S2) is the set of all directed edges e = uv such that u ∈ S1 and v ∈ S2. Then, for any subset S of V(D), E(D)= E(S,S)∪ E(S,S)∪ E(S,S)∪ E(S,S), where S = V(D)− S. For a -voltage assignment on the subgraph S of D, we define a new digraph D×( ,S) as follows. We adjoin an extra element, say ∞, to the group with the property that ∞=∞=∞ for each ∈ ∪{∞}. Notice that ∪ {∞} is a semigroup. The vertex set V(D×( ,S)) is(S × ) ∪ (S × {∞}). For the edges set, we put a directed edge from (u, ) to (v, ) if (i) uv ∈ E(S,S) and = =∞; (ii) uv ∈ E(S,S), , ∈ and (uv) = ; (iii) uv ∈ E(S,S), =∞and ∈ ;or(iv)uv ∈ E(S,S), ∈ and =∞. We call D×( ,S) the derived digraph by a subset S of V(D)and a -voltage assignment on the subgraph S or simply, the derived digraph. In the construction, H.K. Kim, J. Lee / Discrete Mathematics 308 (2008) 555–564 557
Fig. 1. W5 as a Z4-graph with S ={v2}. if the subgraph S has no edge then the voltage assignment is an empty function. But, our construction method still works. Now, we consider a construction method of a -graph G (see Fig. 1). For a -graph G, the digraph G is symmetric and a -digraph. This implies that the quotient graph G/ is symmetric. Hence, to construct a -graph, it suffices to consider the base graph D of our construction as a symmetric digraph. A subset S of V(D)is called a P-subset if the number of directed loops based at each vertex in S = V(D)− S is even. A -voltage assignment on the subgraph S induced by a P-subset S of V(D)is symmetric if
− (i) for each directed loop e based at vi in S, there exists another directed loop e based at vi in S such that (e )= (e) 1 if (e) is not of order 2, and − (ii) for each directed edge e = vivj (i = j) in E(S,S) there exists e = vj vi in E(S,S) such that (e ) = (e) 1.
Lee and Kim obtained the following theorem.
Theorem 1 (Lee and Kim [7], Theorem 2). Let be a finite group and D be a finite connected symmetric digraph. Let S be a subset of V(D) and be a -voltage assignment on the subgraph S of D. Then the derived digraph D×( ,S) = G for some -graph G if and only if S is a P-subset of V(D)and is symmetric.
Now, we consider the adjacency matrix A(D×( ,S)) of the derived digraph D×( ,S). Let be a group and S a P-subset of V(D). Let be a symmetric -voltage assignment on the induced subgraph S of a finite connected digraph D. For each ∈ , let S ( ,) denote the spanning subgraph of the digraph S whose directed edge set is −1 () so that the digraph S is the edge-disjoint union of spanning subgraphs S ( ,), ∈ . In order to define the adjacency matrix of D×( ,S), we define an order relation on the vertex set V(D×( ,S)) of the derived digraph D×( ,S) as follows: let V(D)={v1,v2,...,v|V(D)|} and let S ={v|V(D)|−|S|+1,...,v|V(D)|}. For any two vertices (vi, ) and(vj , ) of D×( ,S), (vi, )(vj , ) if and only if (i) =∞and ∈ , (ii) , ∈ and or (iii) = and i j. For any finite group ,alinear representation of a group over the complex field C is a group homomorphism from to the general linear group GL(r, C) of invertible r × r matrices over C. The number r is called the degree of the representation . For representation theory, we refer to [9]. Suppose that is a finite group. It is clear that P : → GL(||, C) defined by → P(), where P() is the permutation matrix associated with ∈ corresponding to the action of on itself by right-multiplication, is a representation of . It is called the permutation representation. = , , ..., f i Let 1 1 2 be the irreducible representations of and let i be the degree of i for each 1 , where f = f 2 =|| 1 1 and i=1 i . It is well known that the permutation representation P can be decomposed as a direct sum of irreducible representations. In other words, there exists a unitary matrix M of order || such that