Báo cáo toán học: " The tau constant of a metrized graph and its behavior under graph operations" ppsx

The tau constant of a metrized graph and its behavior under graph operations Zubeyir Cinkir ∗ Department of Elementar y Mathematics Teaching Zirve University, Gaziantep, TURKE Y zubeyir.cinkir@zirve.edu.tr Submitted: Sep 13, 2010; Accepted: Mar 27, 2011; Published: Apr 7, 2011 Mathematics Subject Classification: 05C99, 94C99, 05C76 Abstract This paper concerns the tau constant, which is an important invariant of a metrized graph, and which has applications to arithmetic properties of algebraic curves. We give several formulas for the tau constant, and show how it changes under graph operations includin g deletion of an edge, contraction of an ed ge, and union of graphs along one or two points. We show how the tau constant changes when edges of a graph are replaced by arbitrary graphs. We prove Baker and Rumely’s lower bound conjecture on the tau constant for several classes of metrized graphs. 1 Introduction A metrized graph Γ is a finite compact to pological graph equipped with a distance function on their edges. In this paper, we give foundational results on the tau constant τ(Γ), a positive real-valued number. We systematically study τ(Γ), and develop a calculus fo r its computations. Our results in this article and in [3], [4], [5], [6] and [7] are intended to show that τ (Γ) should be considered a fundamental invariant of Γ. Our results extend to weighted graphs. We show that many of the intricate calculations concerning metrized graphs can be obtained in much simpler way by viewing the graph as an electrical circuit and then performing suitable circuit reductions. ∗ I would like to thank Dr. Robert Rumely for his guidance. His continued support and encouragement made this work possible. I would like to thank Dr. Matthew Baker for always being available for useful discussions during and before the preparation of this pap e r. Their suggestions and work were inspiring to me. I also would like to thank to anonymous referee for very helpful and detailed feedback on earlier version of this paper. the electronic journal of combinatorics 18 (2011), #P81 1 One of the motivation to study the tau constant of a metrized graph is the following conjecture of Rumely and Baker: Conjecture 1.1. If ℓ(Γ) =  Γ dx denotes the total length of Γ, then we have inf Γ τ(Γ) ℓ(Γ) > 0, taking the infim um over all metrized graphs Γ with ℓ(Γ) = 0. We call this Baker and Rumely’s lower bound conjecture (see Conjecture 2.13 which is the original form). The tau constant is also closely related to the graph invariants that are the crucial part of Zhang’s conjecture concerning the Effective Bogomolov Conjecture [16]. Recently, Effective Bogomolov Conjecture over function fields of characteristic 0 is proved [6] by relating several graph invariants to the tau constant. We think that this paper presents a self contained background information to understand the tau constant, and the results and the techniques included here will be important to improve the achievements in [6]. In summer 2003 at UGA, an REU group (REU at UGA, in short) lea d by Baker and Rumely studied properties of the ta u constant and the lower bound conjecture. Baker and Rumely [2] introduced a measure valued Laplacian operator ∆ which extends Laplacian operators studied earlier in the articles [8] and [15]. This Laplacian o perator combines the “discrete” Laplacian on a finite gr aph and the “continuous” Laplacian −f ′′ (x)dx on R. Later, Ba ker and Rumely [2] studied harmonic ana lysis on metrized graphs. In terms of spectral theory, the tau constant is the trace of the inverse operator of ∆, acting on functions f for which  Γ fdµ can = 0, when Γ has total length 1. Next, we give a short summary of the results given in this paper. We show how the Laplacian ∆ acts on the product of two functions (see Theorem 2.1): Theorem 1.2. If f and g are C 2 on a metrized graph Γ and both f ′′ (x) and g ′′ (x) are in L 1 (Γ), then ∆ x (f(x)g(x)) = g(x)∆ x f(x) + f(x)∆ x g(x) − 2f ′ (x)g ′ (x)dx. We express the canonical measure µ can on a met rized graph Γ in terms of the voltage function j x (y, z) on Γ (see Theorem 2.8): Theorem 1.3. For any p, q ∈ Γ, 2µ can (x) = ∆ x j x (p, q) + δ q (x) + δ p (x). We give new f ormulas for the tau constant and resistance function (see Theorem 2.18, Lemma 2.20, Theorem 2.21, and Theorem 5.7). For exa mple, we obtain the following results: Theorem 1.4. For any p, q ∈ Γ, τ(Γ) = 1 4  Γ ( d dx j x (p, q)) 2 dx + 1 4 r(p, q). Theorem 1.5. For any p, q ∈ Γ, and for −1 < n ∈ R,  Γ ( d dx j p (x, q)) 2 j p (x, q) n dx = 1 n + 1 r(p, q) n+1 . the electronic journal of combinatorics 18 (2011), #P81 2 Our main focus is to give a systematic study of how the tau constant behaves under common graph operations. We show how the tau constant changes under graph oper- ations such as the deletion of an edge, the contraction of an edge into its end points, identifying any two vertices, and extending or shortening one of the edge lengths of Γ (see Theorem 5.1, Corollary 5.3, Lemma 6.1, Lemma 6.2 and Corollary 7.2). We define a new graph operation which we call “full immersion of a collection o f given graphs into another graph” (see §4), and we show how the tau constant changes under this operatio n. That is, we determine how τ(Γ) behaves under the operation of replacing all edges of a gr aph by copies of suitably normalized graph or collection of graphs (see Theorem 4.7, Theorem 4.9, Theorem 3.4 and Theorem 3.8 ). We prove that the lower bo und conjecture, Conjecture 1.1, holds for several classes of metrized graphs. For complete graphs, we have the following result (see Proposition 2.16): Proposition 1.6. Let Γ be a complete graph on v ≥ 2 vertice s with equal edge lengths. Then we have τ(Γ) ℓ(Γ) =  1 12  1 − 2 v  2 + 2 v 3  . In particular, τ(Γ) ℓ(Γ) ≥ 23 500 , with equality when v = 5. We also include the following unpublished result, due to Baker, with its proof (see Theorem 2.24 and Theorem 2.27): Theorem 1.7. Suppose all edge len gths in a metrized graph Γ are equal. If Γ has v vertices and e edges, then τ(Γ) ℓ(Γ) ≥ 1 12 (1 − v−1 e ) 2 . In particular, Conjecture 1.1 holds with C = 1 108 if w e also have at least 3 edges connected to each ve rtex in Γ. Conjecture 1.1 holds with C = 1 16 for metrized graphs with 2 vertices and any number of edges (see Corollary 2.23 and Proposition 8.3): Proposition 1.8. Le t Γ be a graph with 2 vertices. Then, τ(Γ) ℓ(Γ) ≥ 1 16 , with equality when Γ has 4 edges that have equal edge lengths and have distinct end points. We obtain the following result for any graph whose adjacent vertices are connected by multiple edges (see Theorem 2.2 5): Theorem 1.9. Let Γ be a metrized graph. If any pair of vertices that are connected by an edge are joined by a t l east two edges, then τ(Γ) ℓ(Γ) ≥ 1 48 . That is, Conjecture 1.1 holds with C = 1 48 for such class of metrized graphs. In an another direction, we show the following upper bound for the tau consta nt (see Corollary 5.8, Remark 5.9 and Proposition 2.28): Proposition 1.10. Let Γ be a metrized graph. If Γ can not be disconnected by deleting any of its edges, then τ(Γ) ℓ(Γ) ≤ 1 12 , wi th equality when Γ is a circle graph or one point union of any number of circle graphs. We show that the infimum is not attained by any metrized graph if the lower bound conjecture is true (see Theorem 4.8): the electronic journal of combinatorics 18 (2011), #P81 3 Theorem 1.11. If Conjecture 1.1 is true, then the infimum is not attained by any metrized graph. Moreover, for any metrized graph Γ with small τ (Γ) and ℓ(Γ) = 1, there is a metrized graph β of unit length with τ(β) < τ(Γ). We explicitly compute the tau constant of various metrized graphs esp ecially in §8. We show how our results can be applied to compute the tau constant for various classes of metrized graphs, including those with vertex connectivity 1 or 2. The results here extend those obtained in [3, Sections 2.4, 3.1, 3.2, 3.3, 3 .4 and 3.5]. The tau constant is also related to the Kirchhoff Index of molecular graphs [7]. Metrized graphs appear in many places in arithmetic geometry. R. Rumely [13] used them to develop arithmetic capacity theory, contributing to local intersection theory for algebraic curves over non-archimedean fields. T. Chinburg and Rumely [8] used metrized graphs to define their “capacity pairing”. Another pairing satisfying “desirable” prop- erties is Zhang’s “admissible pairing on curves”, introduced by S. Zhang [15]. Arakelov introduced an intersection pairing at infinity and used analysis on Riemann surfaces to de- rive global results. In the non-archimedean case, metrized graphs appear as the analogue of a Riemann surface. Metrized graphs and their invariants are studied in the articles [15], [16], [10], [3], [4]. Metrized graphs which arise as dual graphs of algebraic curves, and Arakelov Green’s functions g µ (x, y) on the metrized graphs, play an important role in both of the articles [8] and [15]. Chinburg and Rumely worked with a canonical measure µ can of total mass 1 on a metrized graph Γ which is the dua l graph of the special fiber of an algebraic curve C. Similarly, Zhang [15] worked with an “admissible measure” µ ad , a generalization of µ can , of total mass 1 on Γ. The diag onal values g µ can (x, x) ar e consta nt on Γ. M. Baker and Rumely called this constant the “tau constant” of a metrized graph Γ, and denoted it by τ (Γ). Further applications of the results given in this paper can be found in the articles [4], [5], [6] and [7]. 2 The tau constant and the lower bound conjecture In this section, we first recall a few facts abo ut metrized graphs, the canonical measure µ can on a metrized graph Γ, the Laplacian op erator ∆ on Γ, and the tau constant τ (Γ) of Γ. Then we give a new expression for µ can in terms of the voltage function and two arbitrary points p , q in Γ. This enables us to obtain a new formula for the t au constant. We also show how the Laplacian operator ∆ acts on the product of two functions. A metrized graph Γ is a finite connected graph equipped with a distinguished paramet- rization of each of its edges. One can find other definitions of metrized graphs in the articles [2], [15], [1], and the references contained in those articles. A metrized graph can have multiple edges and self-loops. For any given p ∈ Γ, the number of directions emanating from p will be called the valence of p, and will be denoted by υ(p). By definition, there can be only finitely many p ∈ Γ with υ(p) = 2. the electronic journal of combinatorics 18 (2011), #P81 4 For a metrized graph Γ, we denote a vertex set for Γ by V (Γ). We require that V (Γ) be finite and non-empty and that p ∈ V (Γ) for each p ∈ Γ with υ(p) = 2. For a given metrized graph Γ, it is possible to enlarge the vertex set V (Γ) by considering additional points of valence 2 as vertices. For a given graph Γ with vertex set V (Γ), the set of edges of Γ is the set of closed line segments whose end points are adjacent vertices in V (Γ). We denote the set of edges of Γ by E(Γ). Let v := #(V (Γ)) and e := #(E(Γ)). We define the ge nus of Γ to be the first Betti number g := e − v + 1 of the graph Γ. Note that the genus is a topological invariant of Γ. In particular, it is independent of the choice of the vertex set V (Γ). Since Γ is connected, g(Γ) coincides with the cyclomatic number of Γ in combinatorial graph theory. We denote the length of an edge e i ∈ E ( Γ) by L i . The total length of Γ, which will be denoted by ℓ(Γ), is given by ℓ(Γ) = e  i=1 L i . Let Γ be a metrized graph. If we scale each edge of Γ by multiplying its length by 1 ℓ(Γ) , we obtain a new graph which is called normalization of Γ, and will be denoted Γ Norm . Thus, ℓ(Γ Norm ) = 1. We denote the graph obtained from Γ by deletion of the interior points of an edge e i ∈ E(Γ) by Γ− e i . An edge e i of a connected graph Γ is called a bridge if Γ −e i becomes disconnected. If there is no such edge in Γ, it will be called a bridgeles s graph. As in the article [2], Zh(Γ) will be used to denote the set of all continuous functions f : Γ → C such that for some vertex set V (Γ), f is C 2 on Γ\V (Γ) and f ′′ (x) ∈ L 1 (Γ). Let dx be the Lebesgue measure on Γ, and let δ p be the Dirac measure (unit point mass) at p. For v atp, a formal unit vector (a direction) at p, we denote the directional derivative of f at p in the direction of v by d v f(p). That is, d v f(p) = lim t→0 + f(p+tv)−f(p) t . For a function f ∈ Zh(Γ), Baker and Rumely [2] defined the following measure valued Laplacian on a given metr ized graph: ∆ x (f(x)) = −f ′′ (x)dx −  p∈V (Γ)   v at p d v f(p)  δ p (x). (1) See the a r ticle [2] for details and for a description of the largest class of functions f or which a measure valued Laplacian can be defined. We now clarify how the Laplacian operator acts on a product of functions. For any two functions f (x) and g(x) in Zh(Γ), we have f(x)g(x) ∈ Zh(Γ) and ∆ x (f(x)g(x)) = −  f ′′ (x)g(x) + 2f ′ (x)g ′ (x) + f(x)g ′′ (x)  dx −  p∈V (Γ)   v at p (f(p)d v g(p) + g(p)d v f(p)  δ p (x) the electronic journal of combinatorics 18 (2011), #P81 5 = −g(x)f ′′ (x)dx −  p∈V (Γ) g(p)   v at p d v f(p)  δ p (x) − f(x)g ′′ (x)dx −  p∈V (Γ) f(p)   v at p d v g(p)  δ p (x) − 2f ′ (x)g ′ (x)dx = g(x)∆ x f(x) + f(x)∆ x g(x) − 2f ′ (x)g ′ (x)dx. Thus, we have shown the following result: Theorem 2.1. For any f(x) and g(x) ∈ Zh(Γ), we have ∆ x (f(x)g(x)) = g(x)∆ x f(x) + f(x)∆ x g(x) − 2f ′ (x)g ′ (x)dx. The following proposition shows that the Laplacian o n Zh(Γ) is “self -adjoint”, and explains the choice of sign in the definition of ∆. It is proved by a simple integration by parts argument. Proposition 2.2. [15, Lemma 4.a][ 2, Proposition 1.1] For every f, g ∈ Zh(Γ),  Γ g ∆f =  Γ f ∆g, Self-Adjointness of ∆ =  Γ f ′ (x)g ′ (x)dx Green’s Identity. In the article [8], a kernel j z (x, y) giving a fundamental solution of the Laplacian is defined and studied as a function of x, y, z ∈ Γ. For fixed z and y it has the following physical interpretation: when Γ is viewed as a resistive electric circuit with termina ls at z and y, with the resistance in each edge given by its length, then j z (x, y) is the voltage difference between x and z, when unit current enters at y and exits at z (with reference voltage 0 at z). For any x, y, z in Γ, the voltage functio n j x (y, z) on Γ is a symmetric function in y and z, and it satisfies j x (x, z) = 0 and j x (y, y) = r(x, y), where r(x, y) is the resistance function on Γ. For each vertex set V (Γ), j z (x, y) is continuous on Γ as a function of 3 variables. As the physical interpretation suggests, j x (y, z) ≥ 0 fo r a ll x, y, z in Γ. For proofs of these facts, see the articles [8], [2, sec 1.5 and sec 6], and [15, Appendix]. The voltage function j z (x, y) and the resistance function r(x, y) on a metrized graph were also studied by Baker and Faber [1]. Proposition 2.3. [8] For any p, q, x ∈ Γ, ∆ x j p (x, q) = δ q (x) − δ p (x). In [8, Section 2], it was shown that the theory of harmonic functions on metrized graphs is equivalent to the theory of resistive electric circuits with terminals. We now recall the following well known facts from circuit theory. They will be used f r equently and implicitly in this paper and in the papers [4], [5], [6]. The basic principle of circuit analysis is that if one subcircuit of a circuit is replaced by another circuit which has the same resistances between each pair of terminals as the original subcircuit, then all the the electronic journal of combinatorics 18 (2011), #P81 6 p q p q A  B A B  Β s p q A B p q A B A  B  Β Figure 1: Series and Parallel Reductions p q s a b c p q s t b c a  b  c a b a  b  c c a a  b  c  Β p q s p q s t  Β C B A A B  C B  A C A A B  C B  A C B A B  C B  A C C Figure 2: Delta-Wye and Wye-Delta transformations resistances between the terminals of the original circuit are unchanged. The following sub circuit replacements are particularly useful: Series Reduction: Let Γ be a graph with vertex set {p, q, s}. Suppose that p and s are connected by an edge of length A, and that s and q are connected by an edge of length B. Let β be a graph with vertex set {p, q}, where p and q are connected by an edge of length A + B. Then the effective resistance in Γ between p and q is equal to the effective resistance in β between p and q. These are illustrated by the first two graphs in Figure 1. Parallel Reduction: Suppose Γ and β be two graphs with vertex set {p, q}. Suppose p and q in Γ are connected by two edg es of lengths A and B, respectively, and let p and q in β be connected by an edge of length AB A+B (see the la st two graphs in Figure 1). Then the effective resistance in Γ between p and q is equal to the effective resistance in β between p and q. Delta-Wye transformation: Let Γ be a triangular graph with vertices p, q, and s. Then, Γ (with resistance function r Γ ) can be transformed to a Y-shaped graph β ( with resistance function r β ) so that p, q, s become end points in β and the following equivalence of resistances hold: r Γ (p, q) = r β (p, q), r Γ (p, s) = r β (p, s), r Γ (q, s) = r β (q, s). Moreover, for the resistances a, b, c in Γ, we have the resistances bc a+b+c , ac a+b+c , ab a+b+c in β, a s illustrated by the first two graphs in Figure 2. Wye-Delta transformation: This is the inverse Delta-Wye transformation, and is illustrated by the last two graphs in Figure 2. Star-Mesh transformation: An n-star shaped graph ( i.e. n edges with one common point whose other end points are of valence 1) can be transformed into a complet e graph of n vertices (which does not contain the common end point) so that all resistances between the remaining vertices remain unchanged. A more precise description is as follows: Let L 1 , L 2 , · · · , L n be the edges in an n-star shaped graph Γ with common vertex p, the electronic journal of combinatorics 18 (2011), #P81 7 q 1 q 2 q 3 q 4 q 5 q 6 q 1 q 2 q 3 q 4 q 5 q 6 p L 1 L 2 L 3 L 4 L 5 L 6 L 12 L 34 L 45 L 56 L 16 L 13 L 14 L 15 L 26 L 36 L 35 L 25 L 24 L 46 L 23 Figure 3: Star-Mesh transformations when n = 6. where L i is the length of the edge connecting the vertices q i and p (i.e., the resistance between the vertices q i and p. The star-mesh transformation applied to Γ gives a complete graph Γ c on the set of vertices q 1 , q 2 , · · · , q n with n(n−1) 2 edges. The formula for the length L ij of the edge connecting the vertices q i and q j in Γ c is L ij = L i L j · n  k=1 1 L k , for any 1 ≤ i < j ≤ n. When n = 2, the star-mesh transformation is identical to series reduction. When n = 3, the star-mesh transformation is identical to the Wye-Delta transformation, and can be inverted by the Delta-Wye transformation. This is the one case where a mesh can be replaced by a star. When n ≥ 4, there is no inverse transformation for the star-mesh transformation. Figure 3 illustrates the case n = 6. (For more details see the articles [14] or [11]). For any given p and q in Γ, we say that an edge e i is not part of a simple path f r om p to q if a ll walks starting at p, passing through e i , and ending at q must visit some vertex more than once. Another basic principle of circuit reduction is the following tra nsformation: The effective resistances between p and q in both Γ and Γ − e i are the same if e i is not part of a simple path from p to q. Therefore, such an edge e i can be deleted a s far as the resistance between p and q is concerned. For any real- valued, signed Bo r el measure µ on Γ with µ(Γ) = 1 and |µ|(Γ) < ∞, define the function j µ (x, y) =  Γ j ζ (x, y) dµ(ζ). Clearly j µ (x, y) is symmetric, and is jointly continuous in x and y. Chinburg and Rumely [8] discovered that there is a unique real-valued, signed Borel measure µ = µ can such that j µ (x, x) is constant on Γ. The measure µ can is called the canonical measure. Baker and Rumely [2] called the constant 1 2 j µ (x, x) the tau constant of Γ and denoted it by τ (Γ). In terms of spectral theory, as shown in the article [2], the tau constant τ(Γ) is the trace of the inverse of the Laplacian operator on Γ with respect to µ can . The following lemma gives another description of the tau constant. In particular, it implies that the tau constant is positive. Lemma 2.4. [2, Lemma 14.4] For any fixed y in Γ, τ(Γ) = 1 4  Γ  ∂ ∂x r(x, y)  2 dx. the electronic journal of combinatorics 18 (2011), #P81 8 x p q j x p, q j q x, q j p x, q Figure 4: Circuit reduction with reference to 3 points x, p and q. The canonical measure is given by the following explicit formula: Theorem 2.5. [8, Theorem 2.11] Let Γ be a metrized graph . Suppose that L i is the length of edge e i and R i is the effective resistance between the e ndpoints of e i in the graph Γ − e i , when the graph is regarded as an electric circuit with resistances equal to the edge lengths. Then we have µ can (x) =  p∈V (Γ) (1 − 1 2 v(p)) δ p (x) +  e i ∈E(Γ) dx L i + R i , where δ p (x) is the Dirac measure. Corollary 2.6. [2, Corollary 14.2] The measure µ can is the unique measure ν of total mass 1 on Γ maximizing the integral  Γ×Γ r(x, y) dν(x)dν(y). The following theorem expresses µ can in terms of the resistance function: Theorem 2.7. [2, Theorem 14.1] For any p ∈ Γ, the measure µ can (x) is given by µ can (x) = 1 2 ∆ x r(x, p) + δ p (x). It is shown in [8] that as a function of three variables, on each edge j x (p, q) is a quadratic function of p, q, x and possibly with linear t erms in |x − p|, |x − q|, |p − q| if some of p, q, x belong to the same edge. These can be used to show that j x (p, q) is differentiable for x ∈ Γ\  {p, q} ∪ V (Γ)  . Moreover, we have j x (p, q) ∈ Zh(Γ) for each p and q in Γ. For any x, p and q in Γ, we can transform Γ to an Y -shaped graph with the same resistances between x, p, and q as in Γ by applying a sequence of circuit reductions. The resulting graph is shown in Figure 4, with t he corresponding voltage values on each segment. For any given Γ with p, q in V (Γ), and for each x ∈ Γ, the fact that we can transform Γ to a Y -shaped graph as in Figure 4 is a basic result of circuit reductions, and it is a high possibility that this fact is already explained in the literature. Since the circuit reductions giving rise to Figure 4 is fundamental for our later results, we elaborate their details below for reader’s convenience: (i) Start with a metrized graph Γ with p, q in V (Γ), and x ∈ Γ. Recall that Γ is a connected graph. (ii) Enlarge V (Γ) by considering x as a vertex of Γ. the electronic journal of combinatorics 18 (2011), #P81 9 (iii) Apply parallel circuit reduction(s) until the transfo r med graph has no any parallel edges. Note that this procedure if applicable reduces the number of edges. (iv) Apply Series circuit reduction(s) until the transformed graph has no adjacent edges in series connection. Note that this procedure if applicable reduces the number of vertices, and that we don’t apply a series reduction if it deletes any of the fixed vertices p, q, and x. (v) Apply star-mesh transformation to reduce the number of vertices by 1. We want to keep the vertices p, q and x, so any of these vertices can not be the center vertex of the star-mesh transformation. (vi) After a star-mesh transformation, if some parallel edges show up, we apply para llel circuit reduction(s) as in (iii) again, and we apply series reduction(s) as in (iv) again if needed. (vii) We continue applying the reductio ns above until the remaining vertices are only p, q and x. At this stage, the transformed graph is a triangle with vertices p, q and x. (viii) Finally, apply Delta-Wye transformation to obtain a Y -shap ed graph as in Fig ure 4. By Figure 4, we have r(p, x) = j p (x, q) + j x (p, q), r(q, x) = j q (x, p) + j x (p, q), r(p, q) = j q (x, p) + j p (x, q), (2) so ∆ x r(p, x) = ∆ x j p (x, q) + ∆ x j x (p, q), ∆ x r(q, x) = ∆ x j q (x, p) + ∆ x j x (p, q), ∆ x r(p, q) = ∆ x j q (x, p) + ∆ x j p (x, q) = 0. (3) Using these formulas, we can express µ can in terms of the voltage function in the following way: Theorem 2.8. For any p, q ∈ Γ, 2µ can (x) = ∆ x j x (p, q) + δ q (x) + δ p (x). Proof. By Proposition 2.3 and Equation (3), ∆ x r(x, p) = ∆ x j x (p, q) + δ q (x) − δ p (x). (4) Hence, the result follows from Theorem 2.7. Let e i ∈ E(Γ) be an edge for which Γ − e i is connected, and let L i be the length of e i . Suppose p i and q i are the end points of e i , and p ∈ Γ −e i . By applying circuit reductions, Γ − e i can be transformed into a Y -shaped graph wit h the same resista nces between p i , q i , and p as in Γ − e i . The resulting graph is shown by the first gra ph in Figure 5, with the corresponding voltage values on each segment, where ˆ j x (y, z) is the voltage function in Γ − e i . Since Γ − e i has such a circuit reduction, Γ has the circuit reduction shown in the second graph in Figure 5. Throughout this paper, we use the following notation: the electronic journal of combinatorics 18 (2011), #P81 10 [...]... Although Theorem 4.7 is an important progress in this direction, it only provides a partial answer A complete answer is given by Theorem 4.9 below When it is applicable for a metrized graph Γ, Theorem 4.9 reduces both of the computation of the tau constant and Conjecture 2.13 to metrized graphs that are smaller parts of Γ The proof of Theorem 4.7 suggests a further generalization as follows: Let Γ be a graph. .. generalizations of Theorem 3.4 and Theorem 3.8 4 The tau constant and graph immersions In this section, we define another graph operation which will be a generalization of the process of obtaining ΓDA,n from a graph Γ as presented in §3 Let r(x, y) and r n (x, y) be the resistance functions on Γ and ΓDA,n , respectively First we reinterpret the way we constructed ΓDA,n in order to clarify how to generalize... Given a graph Γ and a n-banana graph βn (the graph with n parallel edges of equal length between vertices p and q) we replaced each edge of Γ by βn,i, a copy of βn scaled so that each edge had length nLi Then, we divided each edge length by n2 to have ℓ(ΓDA,n ) = ℓ(Γ) In this operation the following features were important in enabling us to compute τ (ΓDA,n ) in terms of τ (Γ): • We started with a graph. .. will be clear when we examine its relation to τ (Γ) in later sections In some sense it is the basic hard-toevaluate graph integral, and many other integrals can be evaluated in terms of it Remark 4.1 (Scaling Property for Ap,q,Γ) Let Γ be a graph and let β be a graph obtained by multiplying length of each edge in E(Γ) by a constant c If x on an edge ei in Γ separates the edge ei into two parts with... We call such a graph a bouquet graph When e = 2, Γ is just a union of two circles along p Proposition 2.28 Let Γ be a bouquet graph Then we have τ (Γ) = ℓ(Γ) 12 Proof The result follows from Corollary 2.17 and the additive property of the tau constant 3 The tau constants of metrized graphs with multiple edges Let Γ be an arbitrary graph; write E(Γ) = {e1 , e2 , , ee } As before, let Li be the length... of an edge ej of a graph Γ when the edge ej is deleted from Γ Figure 6 shows the edge replacement for an edge when n = 4 A graph with two vertices and m edges connecting the vertices will be called a m-banana graph Lemma 3.1 Let β be a m-banana graph, as shown in Figure 7, such that Li = L for each ei ∈ β Let r(x, y) be the resistance function in β, and let p and q be the end points L of all edges Then,... universal constant C > 0 such that for all metrized graphs Γ, τ (Γ) ≥ C · ℓ(Γ) Remark 2.14 Based on results and examples given later in the paper, it seems probable 1 that one can take C = 108 Remark 2.15 [2, pg 265] If we multiply all lengths on Γ by a positive constant c, we obtain a graph Γ′ of total length c · ℓ(Γ) Then τ (Γ′ ) = c · τ (Γ) This will be called the scale-independence of the tau constant. .. i Proof By Theorem 2.21, τ (Γ − ei ) = 4 Γ−ei ( dx jx (pi , qi ))2 dx + the formula of Corollary 5.3, one obtains the result Ri 4 Substituting this into Note that Corollary 5.4 shows that the tau constant τ (Γ) approaches ℓ(Γ) (the tau 12 constant of a circle graph) as we increase one of the edge lengths and fix the other edge lengths One wonders how τ (Γ) changes if one changes the length of an edge... Corollary 2.19 the electronic journal of combinatorics 18 (2011), #P81 25 e1 a e3 b e4 e5 p e2 q Figure 11: Diamond graph Example 4.4 Let Γ be the graph, which we call the “diamond graph , shown in Figure 11 Assume the edges {e1 , , e5 } and the vertices {a, b, p, q} are labeled as shown Let each edge length be L By the symmetry of the graph, edges e1 , e2 , e3 and e4 make the same contribution to Ap,q,Γ... r(p, y) for each x ∈ Γ1 and y ∈ Γ2 By using this fact and Corollary 2.4, we obtain τ (Γ1 ∪ Γ2 ) = τ (Γ1 ) + τ (Γ2 ), which we call the “additive property” of the tau constant It was initially noted in the REU at UGA The following corollary of Theorem 2.21 was given in [2, Equation 14.3] Corollary 2.22 Let Γ be a tree, i.e a graph without cycles Then, τ (Γ) = the electronic journal of combinatorics 18 . of an edge, contraction of an ed ge, and union of graphs along one or two points. We show how the tau constant changes when edges of a graph are replaced by arbitrary graphs. We prove Baker and Rumely’s. of how the tau constant behaves under common graph operations. We show how the tau constant changes under graph oper- ations such as the deletion of an edge, the contraction of an edge into its. analogue of a Riemann surface. Metrized graphs and their invariants are studied in the articles [15], [16], [10], [3], [4]. Metrized graphs which arise as dual graphs of algebraic curves, and Arakelov