A note on the complexity of Graphs - Faculty web pages

A NOTE ON THE COMPLEXITY OF GRAPHSSam NorthshieldSUNY-College at Plattsburgh and University of MinnesotaAbstract. The number of splanning trees in a finite graph is first expressed asthe derivative (at 1) of a determinant and then in terms of a zeta function. Thisgeneralizes a result of Hashimoto to non-regular graphs.Let G be a finite graph. The complexity of G, de<strong>note</strong>d κ, is the number ofspanning trees in G. This quantity has long been known to be related to matricesassociated with G (see [1]). When G is regular, Hashimoto [2] expressed κ as alimit involving the zeta function of the graph and asked if his expression still holdsfor irregular graphs. In this <strong>note</strong>, we show that the answer is yes. In particular,we derive a formula for the complexity as the derivative of a determinant involvingthe adjacency matrix of the graph and use this and a generalized version of Ihara’stheorem ([3,4]) to get the desired result.Let G be a graph with ν vertices and ε edges. Suppose we order the vertices:(x 1 , . . . , x ν ). The adjacency matrix A = (a ij ) is the matrix of zeros and ones suchthat a ij = 1 if and only if x i and x j are adjacent. We let D = (d ij ) be the diagonalelement with d ii = d i − 1 where d i is the degree of the vertex x i . Finally, letQ = D − I. For a complex variable u, letTheorem. f ′ (1) = 2(ɛ − ν)κ.f(u) = det(I − uA + u 2 Q).Proof. Let M u = I − uA + u 2 Q and let M u k = (mu k;ij ) de<strong>note</strong> the matrix M u witheach entry of the k th row replaced by its corresponding derivative with respect tou. Thendet(M u ) ′ = ∑ π= ∑ jsgn(π)( ∏ i∑sgn(π) ∏πim u iπ(i) )′m u j;iπ(i) = ∑ jdet(M u j ).Note that m 1 k;ij = d iδ ij − a ij + (d k − 2)δ ki δ ij , and thusf ′ (1) = ∑ kdet(M + (d k − 2)R k )This <strong>note</strong> was written while the author was on sabbatical leave at the University of Minnesota.The author appreciated the hospitality of the math department during that time.1Typeset by AMS-TEX

2 SAM NORTHSHIELDwhere M = M 1 and R k = (r k;ij ) is defined by r k:ij = δ ki δ ij .In general,if ˜m ij is the cofactor of m ij in M, thendet(M + cR k ) = ∑ j(M + cR k ) kj (−1) j+k ˜m kj= ∑ jm kj (−1) j+k ˜m kj + c ˜m kk = c ˜m kk(since det(M) = 0). By Biggs ([1], theorem 6.3), ˜m ij = κ for all i and j and thusf ′ (1) = κ ∑ k (d k − 2) = 2(ɛ − ν)κ. □We define the zeta function of G to beZ(u) = ∏ i(1 − u ω i) −1where (ω 1 , ω 2 , . . . ) are the lengths of prime cycles in G (see [3] or [4] for definitions).A generalization of a theorem of Ihara’s is thatZ(u) =1(1 − u 2 ) ɛ−ν f(u)(see [3] or [4]). The following generalization of [2; theorem 7.7] is immediate.Corollary. lim u→1 − Z(u)(1 − u) ɛ−ν+1 = − 2−ɛ+ν−1(ɛ−ν)κ .References1. N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1974.2. K. Hashimoto, Zeta Functions of Finite Graphs and Representations of p-Adic Groups, AdvancedStudy in Pure Math., vol. 15, Academic Press, NY, 1989, pp. 211-280.3. S. Northshield, Several proofs of Ihara’s theorem, IMA Preprint Series no. 1459.4. H.M. Stark and A.A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121(1996), no. 1, 124–165.