The Geometric Spectrum of a Graph and Associated Curvatures

Abstract

We approach the problem of defining curvature on a graph by attempting to attach a ‘best-fit polytope’ to each vertex, or more precisely what we refer to as a configured star. How this should be done depends upon the global structure of the graph which is reflected in its geometric spectrum. Various curvatures naturally arise from local liftings of the graph into a suitable Euclidean space.

Appendices

Appendix 1: The Geometric Spectrum, Gröbner Bases and the γ-Polynomial

To compute the geometric spectrum, even for simple graphs, is quite challenging. We consider some fundamental cases and then make a simplifying assumption in order to apply the technique of Gröbner bases. This motivates the construction of a new polynomial invariant associated to a graph.

7.1.1 First Cases

Consider a vertex v ₀ of degree 2. Suppose that a solution ϕ to (7.2) is non-constant on a neighborhood of this vertex, thus ϕ takes a different value on at least one of its two neighbors v ₁, v ₂, say v ₁. By the normalization (7.3), we may suppose that ϕ(v ₀) = 0 and that ϕ(v ₁) = 1. If we let ϕ(v ₂) = z as illustrated in Fig. 7.9, then at vertex v ₀, Eq. (7.2) takes the form:

$$\displaystyle{ \frac{\gamma } {2}(1 + z)^{2} = 1 + z^{2}\,. }$$

Fig. 7.9

The general invariant star at a vertex of degree 2

Suppose that z ≠ − 1. Then the requirement that γ be real is equivalent to

$$\displaystyle{ \mathrm{either}\quad \mathfrak{I}(z) = 0\quad \mathrm{or}\quad \vert z\vert = 1. }$$

If z is real and z ≠ ± 1, then γ > 1, so we reject this case since γ will not lie in the geometric spectrum. Suppose that z is not real and write it in polar coordinates: z = e^iα. Then

$$\displaystyle{ \gamma = \frac{2\cos \alpha } {1+\cos \alpha } = \frac{2\cos \theta } {\cos \theta -1}\,, }$$

where θ = π −α is the external angle. The two limiting cases α = 0 and α = π correspond to γ = 1 and γ = −∞, respectively. This justifies the admissibility of the value 1 in the geometric spectrum at a vertex of degree 2 (see Sect. 7.3.2).

As a consequence, for a cyclic graph, solutions to (7.2) correspond to realizations in the plane as a polygonal framework with sides of equal length. A cyclic graph on three vertices can only have one such realization (up to similarity transformation) as an equilateral triangle, so that γ = 2∕3 is the only value in the geometric spectrum.

A cyclic graph on four vertices is realized as a rhombus, with the possibility of its edges collapsing onto themselves as indicated in Fig. 7.10.

Fig. 7.10

An invariant cyclic graph of order 4 has geometric spectrum corresponding to realizations in the plane as a polygonal framework with sides of equal length

In this case, the geometric spectrum has continuous components corresponding to continuous deformations of the rhombus which may be parametrized by one of the internal angles. Branching phenomena occurs as edges collapse onto themselves. As the number of vertices increases, so too does the number of variables that parametrize the geometric spectrum.

Let us turn to the other extreme, that of a complete graph. The complete graph on three vertices is cyclic and as shown above, there is just one element in its geometric spectrum, the constant value γ = 2∕3. Since any invariant star in $\mathbb{R}^{3}$ with three external vertices is necessarily configured [3], it follows that a complete graph on four vertices which has a realization as an invariant framework in $\mathbb{R}^{3}$ is necessarily a tetrahedron with γ taking the constant value 3∕4. However, the results of Sect. 7.3 only guarantee a local lifting of a vertex and its neighbors in an invariant way, so in general, we don’t know if such a realization exists. However, we can use a computer to solve the equations for a sufficiently small number of vertices.

Consider the complete graph on N + 1 vertices. Label the vertices by the integers 0, 1, …, N and set γ _j = γ( j), z _j = ϕ( j). After normalization (7.3), we may suppose that z ₀ = 0 and z ₁ = 1. Then (7.2) becomes:

$$\displaystyle{ \frac{\gamma _{k}} {N}\left (\sum _{j=0}^{N}(z_{ k} - z_{j})\right )^{2} =\sum _{ j=0}^{N}(z_{ k} - z_{j})^{2}\quad (k = 0,1,2,\ldots,N)\,, }$$

with the constraints that each γ _k be real and < 1. The software MAPLE can now solve this algebraic system at least up to N = 5 (the complete graph on 6 vertices), which confirms that the only element in the geometric spectrum is given by γ constant equal to N∕(N + 1). We currently don’t have a mathematical (non-computer) proof of this fact and don’t know if the constancy of γ persists for N > 5.

7.1.2 The Constant Geometric Spectrum and the γ-Polynomial

From now on, to simplify matters, we consider only constant values of γ that may lie in the spectrum and attempt an algebraic geometric approach to compute this set which we refer to as the constant geometric spectrum.

Let G = (V, E) be a connected graph. We are interested in the possible real numbers γ for which there are non-constant solutions to the equation:

$$\displaystyle{ \gamma \varDelta \phi ^{2} = (d\phi )^{2}\,. }$$

(7.27)

Any solution is invariant by ϕ ↦ λϕ + μ, for complex constants λ, μ(λ ≠ 0). We will no longer insist that γ < 1. Consider first how to parametrize all possible complex fields on the graph under this invariance.

Label the vertices of the graph x ₁, x ₂, …, x _N and consider a non-constant complex field ϕ that assigns the value ϕ(x _k ) = z _k to vertex x _k . Then the space of all such fields is identified with the complex space $\mathbb{C}^{N}\setminus \{\mu (1,1,\ldots,1):\mu \in \mathbb{C}\}$. Up to the equivalence (z ₁, …, z _N ) ∼ (z ₁ + μ, …, z _N + μ), we can identify these fields with the set Π∖{0}, where Π is the linear subspace $\varPi =\{ \mathbf{Z} = (z_{1},\ldots,z_{N}) \in \mathbb{C}^{N}: z_{1} + \cdots + z_{n} = 0\} \subset \mathbb{C}^{N}$. In effect, given any non-constant field (z ₁, …, z _N ), then (z ₁ + μ, …, z _N + μ) lies in the plane z ₁ + ⋯ + z _N = 0, when we set $\mu = -\frac{1} {N}(z_{1} + \cdots + z_{N})$. By non-constancy, this is non-zero. Furthermore, it is clear that any two equivalent fields correspond to the same point.

Now consider the relation Z ∼ λ Z, for $\lambda \in \mathbb{C}\setminus \{0\}$. This defines the moduli space of fields up to equivalence to be $\mathcal{Z}:= \mathbb{C}\mathrm{P}^{N-2}$. Specifically, given a point $[z_{1},\ldots,z_{N-1}] \in \mathcal{Z}$ in homogeneous coordinates, we define a representative field by $(z_{1},\ldots,z_{N-1},z_{N} = -\sum _{k=1}^{N-1}z_{k}) \in \mathbb{C}^{N}$. In practice, we can set a field equal to 0 and 1 on two selected vertices x ₀ and x ₁, respectively, and label the other vertices arbitrarily. This is only one chart and we miss those fields which coincide at these two vertices.

If we consider γ as an arbitrary complex parameter, then (7.27) imposes a constraint at each vertex, so we have N equations in N − 1 parameters. In general these are independent so that this is an overdetermined system, which may have no solutions.

For each ℓ = 2, …, N, consider the following set of N polynomials defined over the algebraically closed field $\mathbb{C}$. The variables are the values {z ₁, …, z _N } of a field on G with constraints z ₁ = 0 and z _ℓ = 1; we suppose the degree of vertex j is d _j and that z _jk ∈ {z ₁, …, z _N } (k = 1, …, d _j ) are the values of the field on the neighbors x _jk of x _j . The polynomials are then defined by

$$\displaystyle{ f_{j}^{\ell}:= \frac{\gamma } {d_{j}}\left (\sum _{k=1}^{d_{j} }(z_{j} - z_{jk})\right )^{2} -\sum _{ k=1}^{d_{j} }(z_{j} - z_{jk})^{2}\qquad (z_{ 1} = 0,\,z_{\ell} = 1)\,, }$$

in the N − 1 complex variables $\{\gamma,z_{2},z_{3},\ldots,\widehat{z_{\ell}},\ldots,z_{N}\}$. Recall some facts and terminology from commutative algebra. We are particularly interested in the techniques of Gröbner bases, for which we refer the reader to [1, 27].

For an ideal I = < f ₁, …, f _N > in a polynomial ring $\mathbb{C}[x_{1},x_{2},\ldots,x_{M}]$, we denote by V (I) the corresponding variety: f ₁ = 0, f ₂ = 0, …, f _N = 0. Then I is said to be zero-dimensional if V (I) is finite. A Gröbner basis for I is a basis of polynomials which can be constructed from f ₁, …, f _N using a particular algorithm, called the Buchberger algorithm. To employ this algorithm, one is required first to choose an order on monomials. We shall only be concerned with lexicographical order here, which means we first choose an ordering of the variables, say x ₁ > x ₂ > ⋯ > x _M and then order monomials $x^{\alpha }:= x_{1}^{\alpha _{1}}\cdots x_{M}^{\alpha _{M}}$, $x^{\beta }:= x_{1}^{\beta _{1}}\cdots x_{M}^{\beta _{M}}$, by x ^α < x ^β if and only if the first coordinate from the left for which α _i and β _i are different, satisfies α _i < β _i . With respect to the monomial order, every polynomial f in I has a leading term lt ( f) which is the product lt ( f) = lc ( f)lm ( f) of the leading coefficient with the leading monomial.

A set of non-zero polynomials G = {g ₁, …, g _P } in I is called a Gröbner basis for I if and only if for all f ∈ I such that f ≠ 0, there is a g _j in G such that lm (g _j ) divides lm ( f). The Gröbner basis is further called reduced if for all j, lc (g _j ) = 1 and g _j is reduced with respect to G∖{g _j }, that is, no non-zero term in g _j is divisible by any lm (g _k ) for any k ≠ j. A theorem of Buchberger states that every non-zero ideal has a unique reduced Gröbner basis with respect to a monomial order [9]. Gröbner bases are particularly useful for understanding the solution set of a system of polynomial equations.

Let I be an ideal in the polynomial ring $\mathbb{C}[x_{1},x_{2},\ldots,x_{M}]$ and let G = {g ₁, …, g _P } be the unique reduced Gröbner basis with respect to the lexicographical ordering induced by the order x ₁ > x ₂ > ⋯ > x _M . Then V (I) is finite if and only if for each j = 1, …, M, there exists a g _k ∈ G such that $\mathrm{lm}\,g_{k} = x_{j}^{n_{j}}$ for some natural number n _j . As a consequence, if I is a zero-dimensional ideal, it follows that we can order g ₁, …, g _P so that g ₁ contains only the variable x _M , g ₂ contains only x _M , x _M−1 and so on. This is because the leading monomial of one element, g ₁ say, of G must be a power of x _M and then no other term of g ₁ can contain powers of any other variable (for such terms would be greater that any power of x _M with respect to the monomial order), and so on for successive elements g ₂, g ₃, … of G. We note also that V (I) is empty if and only if 1 ∈ G.

It is also the case that, with the above hypothesis, the polynomial g ₁ is the least degree univariate polynomial in x _M which belongs to I (any zero-dimensional ideal contains such a polynomial for every variable). For if there was another univariate polynomial p(x _M ) with deg p < deg g ₁, then lm p would divide lm g ₁ in a strict sense, which would contradict the fact that G is a reduced Gröbner basis. Let us now return to the case under consideration.

For each ℓ = 2, …, N, consider the ideal I _ℓ = < f ₁ ^ℓ, …, f _N ^ℓ >. Suppose that for each ℓ = 2, …, N this admits a least degree univariate polynomial p _ℓ in γ. This can be constructed by first choosing a lexicographical ordering of the variables with γ the smallest and then applying an algorithm (say the Buchberger algorithm) to construct the unique reduced Gröbner basis for I _ℓ . The first element of this basis gives p _ℓ .

Definition 7.6.

We define the γ-polynomial p = p _G of the connected finite graph G = (V, E) to be the least common multiple of the least degree univariate polynomials p _ℓ (ℓ = 2, …, N) in γ associated to the Eq. (7.2) for fields (z ₁, …, z _N ) on G with z ₁ = 0 and z _ℓ = 1:

$$\displaystyle{ p:= \mathrm{lcm}\,(\,p_{2},\ldots,p_{N})\,, }$$

when each p _ℓ exists.

The γ-polynomial p(γ) is defined up to rational multiple and has rational coefficients. This is because the initial polynomials f _j ^ℓ used to define p all have integer coefficients and the Buchberger algorithm then generates polynomials with rational coefficients—it involves at most division by coefficients—see [1]. Clearly p depends only on the isomorphism class of a graph and in the case when the Eq. (7.27) admit no solutions for γ constant and complex, then p ≡ 1. In this case we shall say that p is trivial. The polynomials p _ℓ and so p may still be well-defined even if the solution set of the equations is infinite (that is the corresponding ideal is no longer zero-dimensional). In fact we know of no case when they are not well-defined.

The constant geometric spectrum arises as real roots of p (the problem of establishing the discreteness of the spectrum is clearly intimately related to knowing if p is well-defined in all cases). However, not all real roots may occur in the spectrum, for in general they must also solve the other equations determined by the Gröbner basis: g ₁ = 0, …, g _P = 0. Examples below illustrate this property. We know of no two non-isomorphic connected graphs with non-trivial γ-polynomial having the same γ-polynomial.

The examples of the triangle C ₃ (the cyclic graph on three vertices) and the bipartite graphs K ₂₃ and K ₃₃ are instructive. We label the vertices as shown in Fig. 7.11 and consider fields ϕ taking the values ϕ(x _j ) = z _j at each vertex x _j .

Fig. 7.11

Three graphs for which the γ-polynomial can be easily calculated by an appropriate labeling of the vertices

For the triangle, there are precisely two solutions to (7.27) when we normalize so that z ₁ = 0, z ₂ = 1; specifically $z_{3} = \frac{1} {2} \pm \mathrm{i}\frac{\sqrt{3}} {2}$. Then p = p ₂ = p ₃ = 3γ − 2 is the γ polynomial and the only constant element (in fact the only element) of the geometric spectrum is the unique root γ = 2∕3.

For K ₂₃, we find p ₂ = 1 with no solution and p ₃ = γ ² − 2γ + 1 = (γ − 1)² with solution z ₁ = 0, z ₃ = 1, z ₂ = 0, z ₅ = λ arbitrary and $z_{4} = [1 +\lambda \pm \sqrt{3}(1-\lambda )\mathrm{i}]/2$. Then p = γ ² − 2γ + 1 and the constant geometric spectrum is given by Σ = {1}.

For K ₃₃, we find p ₂ = 9γ ² − 26γ + 17 = (γ − 1)(9γ − 17) and p ₄ = 9γ ³ − 35γ ² + 43γ − 17 = (γ − 1)p ₂, so that the γ-polynomial p = 9γ ³ − 35γ ² + 43γ − 17. Although this has γ = 17∕9 as a root, the constant geometric spectrum Σ = {1}. In fact for γ = 1, z ₁ = 0, z ₂ = 1 we find a two complex parameter family of solutions to (7.27). The next example shows that even for simple graphs, the γ-polynomial can be quite complicated.

Consider the graph of constant degree three on six vertices whose edges form two concentric triangles as shown in Fig. 7.12.

Fig. 7.12

An example of a simple graph with complicated γ-polynomial

The γ-polynomial is given by

$$\displaystyle{ \begin{array}{l} 5859375\gamma ^{10} - 67656250\gamma ^{9} + 333521875\gamma ^{8} - 926025000\gamma ^{7} \\ \qquad \qquad \;\;\ \, + 1603978830\gamma ^{6} - 1808486028\gamma ^{5} + 1339655598\gamma ^{4} \\ \qquad \qquad \quad - 639892872\gamma ^{3} + 186760323\gamma ^{2} - 29598858\gamma + 1883007\,. \end{array} }$$

This has eight real roots, four of which are rational: γ = 3∕5, 21∕25, 1, 3. There remain two conjugate complex roots. Since the γ-polynomial differs from that of the bipartite graph K ₃₃ calculated above, we deduce that these two graphs of constant degree three on six vertices cannot be isomorphic.

It is interesting to consider the real roots that are ≥ 1, which can be seen to arise from surjective mappings $\phi: V \rightarrow S_{N}:=\{ z_{0},\ldots,z_{N}\} \subset \mathbb{C}$, where z ₀, …, z _N are the images under any orthogonal projection to the complex plane of the vertices of a regular N-simplex in $\mathbb{R}^{N}$. This is because of the translation-invariant relation

$$\displaystyle{ (z_{0} + \cdots + z_{N})^{2} = (N + 1)(z_{ 0}^{2} + \cdots + z_{ N}^{2}) }$$

between these projections whenever N ≥ 2 [13]. For example, if N = 4, we may take S ₄ = {0, 1, i, 1 + i}. When N = 1, we take S ₁ = {0, 1} and (z ₀ + z ₁)² = z ₀ ² + z ₁ ².

The root 3 corresponds to the solution ϕ: V → S ₁ = {0, 1} to (7.2) which takes the value 0 say, on the vertices of the inner triangle and 1 on the vertices of the outer triangle. The root 1 corresponds to the solution $\phi: V \rightarrow S_{2} =\{ 0,1, \frac{1} {2} + \mathrm{i}\frac{\sqrt{3}} {2} \}$, as indicated in Fig. 7.13. Thus, ϕ takes on the colors red, green and blue which are in bijective correspondence with S ₂.

Fig. 7.13

Certain roots of the γ-polynomial correspond to graph colorings

More generally, if we can “color” the vertices of a regular graph G of degree d with S _N in such a way that each vertex is connected by an edge to precisely one vertex of each of the other colors, then γ = d∕(N + 1) will be an element of the constant geometric spectrum. Clearly we must have d ≥ N. For the complete graph on N + 1 vertices, we can take the coloring given by any bijection V → S _N to give γ = N∕(N + 1). The above example can be generalized by taking two copies of the complete graph on N + 1 vertices and connecting each vertex of one of the graphs to precisely one vertex of the other in a bijective correspondence. Now d = N + 1 and γ = 1 lies in the constant geometric spectrum. Relations to vertex colorings suggest potential connections between the γ-polynomial and other more well-known polynomial invariants, such as the Tutte polynomial.

Finally, let us consider the γ-polynomial of the complete graph on N + 1 vertices. We may apply the method of Gröbner bases with an appropriate lexicographical ordering which produces the γ-polynomial as its first basis element. In fact, by the symmetry of the complete graph, it is clear that all the univariate polynomials p _ℓ (ℓ = 2, …, N) in Definition 7.6 are identical. If we denote the γ-polynomial by p(γ), then using MAPLE, we obtain the suggestive list given in Table 7.3.

Table 7.3 The γ-polynomial for the complete graph on N + 1 vertices

It is reasonable to conjecture that for N even, the γ-polynomial is given by p(γ) = (N + 1)γ − N and that for N = 2k + 1 odd, it is given by

$$\displaystyle\begin{array}{rcl} p(\gamma )& =& 2(k + 1)^{2}\gamma ^{2} - 3(k + 1)(2k + 1)\gamma + (2k + 1)^{2} {}\\ & =& \Big(2(k + 1)\gamma - (2k + 1)\Big)\Big((k + 1)\gamma - (2k + 1)\Big)\,. {}\\ \end{array}$$

Appendix 2: New Solutions from Old

7.2.1 Holomorphic Mappings Between Graphs

The natural class of mappings between graphs which preserve Eq. (7.2) are the so-called holomorphic mappings. These were introduced for simple graphs under the name semi-conformal mappings by Urakawa [28], as the class of maps which preserve local harmonic functions (i.e. functions which are harmonic at a vertex). The notion was later extended to non-simple graphs by Baker and Norine [5, 6], who used the term holomorphic mapping. In [4], it was shown that the holomorphic mappings are precisely the class of mappings which preserve local holomorphic functions, defined as solutions to (7.2) for which γ ≡ 0. The definition requires that we restrict to mappings of graphs that determine a well-defined mapping of the tangent space at each vertex, which also justifies our inclusion of the zero vector in the definition of tangent space (see Sect. 7.2).

Definition 7.7.

Let f: G = (V, E) → H = (W, F) be a mapping between graphs. Then f is holomorphic if

• (i) x ∼ y implies either f(x) = f( y) or f(x) ∼ f( y);

• (ii) there exists a function $\lambda: V \rightarrow \mathbb{N}$ such that for all x ∈ V and for all z ^′ ∼ z = f(x), we have

  $$\displaystyle{ \lambda (x) =\lambda (x,z^{{\prime}}) = \sharp \{x^{{\prime}}\in V: x^{{\prime}}\sim x,f(x^{{\prime}}) = z^{{\prime}}\}\,, }$$

  is independent of the choice of z ^′; we set λ(x) = 0 if f(x ^′) = z for all x ^′ ∼ x. Call λ the dilation of f.

Proposition 7.6.

Let f: G = (V, E) → H = (W, F) be a holomorphic mapping between graphs of dilation $\lambda: V \rightarrow \mathbb{N}$ . Suppose $\psi: W \rightarrow \mathbb{C}$ satisfies the equation

$$\displaystyle{ \mu (\varDelta \psi )^{2} = (\mathrm{d}\psi )^{2}\,, }$$

for some $\mu: W \rightarrow \mathbb{R}$ . Then for each x ∈ V such that λ(x) ≠ 0, the function ϕ = ψ ∘ f satisfies (7.2) at x with

$$\displaystyle{ \gamma (x) = \frac{d_{x}\mu (\,f(x))} {\lambda (x)d_{f(x)}} \,, }$$

(7.28)

where d _z also denotes the degree of a vertex z ∈ W.

Proof.

Let f: G = (V, E) → H = (W, F) be a holomorphic mapping between graphs of dilation $\lambda: V \rightarrow \mathbb{N}$. Let x ∈ V and set z = f(x). Then

$$\displaystyle{ \frac{\mu (z)} {d_{z}} \left (\sum _{z^{{\prime}}\sim z}(\psi (z^{{\prime}}) -\psi (z))\right )^{2} =\sum _{ z^{{\prime}}\sim z}(\psi (z^{{\prime}}) -\psi (z))^{2}. }$$

Since f is holomorphic

$$\displaystyle{ \sum _{x^{{\prime}}\sim x}[(\psi \circ f)(x^{{\prime}}) - (\psi \circ f)(x)] =\lambda (x)\sum _{ z^{{\prime}}\sim z}(\psi (z^{{\prime}}) -\psi (z))\,. }$$

Suppose that λ(x) ≠ 0. Then

$$\displaystyle\begin{array}{rcl} \sum _{x^{{\prime}}\sim x}[(\psi \circ f)(x^{{\prime}}) - (\psi \circ f)(x)]^{2}& =& \lambda (x)\sum _{ z^{{\prime}}\sim z}(\psi (z^{{\prime}}) -\psi (z))^{2} {}\\ & =& \frac{\lambda (x)\mu (z)} {d_{z}} \left (\sum _{z^{{\prime}}\sim z}(\psi (z^{{\prime}}) -\psi (z))\right )^{2} {}\\ & =& \frac{\mu (z)} {\lambda (x)d_{z}}\left (\sum _{x^{{\prime}}\sim x}[(\psi \circ f)(x^{{\prime}}) - (\psi \circ f)(x)]\right )^{2}\,, {}\\ \end{array}$$

from which the formula follows. If on the other hand λ(x) = 0, then f(x ^′) = f(x) for all x ^′ ∼ x and both sides of (7.2) vanish. □

Given a holomorphic mapping between graphs, the above proposition shows how an element of the geometric spectrum on the co-domain determines one on the domain. A simple example of a holomorphic mapping between planar graphs is given in Fig. 7.14. In this example, the outer “wheel” of the domain graph is mapped cyclically onto the outer wheel of the image, covering it twice, while the central vertex of the domain is mapped onto the central vertex of the image.

Fig. 7.14

An example of a holomorphic mapping between graphs: the outer wheel covers its image twice, while the central vertex is preserved

The dilation at the central vertex is given by λ = 2, whereas it equals 1 at the other vertices. With reference to Example 7.2, we see that the constant value γ ≡ 1∕3 is also an element of the geometric spectrum of the domain graph, the doubling of the degree at the central vertex being exactly compensated for by the doubling of the dilation in the formula (7.28).

7.2.2 Applying Normalization to Construct New Solutions

Another method to construct new solutions to (7.2) is to exploit the freedom to normalize a solution ϕ by (7.3). Whilst not exhaustive, we describe some examples.

The process of collapsing was described at the end of Sect. 7.4, whereby edges that connect vertices on which a solution to (7.2) takes on the same value can be removed. This operation can be reversed as follows. Given a non-constant solution ϕ ₁ to (7.2) on a graph G ₁ = (V ₁, E ₁), then for two vertices x ₁, y ₁ ∈ V ₁ where ϕ ₁(x ₁) ≠ ϕ ₁( y ₁), we can apply the normalization (7.3) and suppose that ϕ ₁(x ₁) = 0 and ϕ ₁( y ₁) = 1. Similarly for a non-constant solution ϕ ₂ to (7.2) on a graph G ₂ = (V ₂, E ₂), we may find two vertices x ₂, y ₂ and normalize so that ϕ ₂(x ₂) = 0 and ϕ ₂( y ₂) = 1. As illustrated in Fig. 7.15, define a new graph G whose vertex set V = V ₁ ∪ V ₂ and whose edge set $E = E_{1} \cup E_{2} \cup \{\overline{x_{1}x_{2}},\overline{y_{1}y_{2}}\}$.

Fig. 7.15

Solutions to equation (7.2) on two graphs can be normalized at two vertices in such a way that the graphs can be connected by edges to give a new solution

Note that we are not obliged to add both edges, and indeed we can connect any vertices on which ϕ ₁ and ϕ ₂ take on the same value. We then define $\phi: V \rightarrow \mathbb{C}$ by ϕ(x) = ϕ ₁(x) if x ∈ V ₁ or ϕ(x) = ϕ ₂(x) if x ∈ V ₂. Clearly ϕ satisfies (7.2), but with γ modified to take into account the fact that the degrees at x ₁, y ₁, x ₂, y ₂ have increased by one.

As a variant, if both x ₁ and y ₁ and x ₂ and y ₂ are connected by edges in G ₁ and G ₂, respectively, then we can remove these edges and replace them by $\overline{x_{1}y_{2}}$ and $\overline{y_{1}x_{2}}$. Thus the new graph G = (V, E) (illustrated in Fig. 7.16) has V = V ₁ ∪ V ₂ and $E = (E_{1}\setminus \{\overline{x_{1}y_{1}}\}) \cup (E_{2}\setminus \{\overline{x_{2}y_{2}}\}) \cup \{\overline{x_{1}y_{2}},\overline{y_{1}x_{2}}\}$. As before we can define a solution to (7.2) to be the restriction of ϕ ₁ to V ₁ and ϕ ₂ to V ₂, but now the degrees are preserved, so γ is unchanged.

Fig. 7.16

Solutions to equation (7.2) on two graphs can be normalized at two vertices to allow edge rotations which connect the two graphs, giving a new solution

7.2.3 The Preferential Attachment Model

In random graph theory, an important generative model is the preferential attachment scheme which proceeds as follows. For the current graph G, add a new vertex y and add an edge $\overline{xy}$ from y by randomly and independently choosing x in proportion to the degree of x in G. As demonstrated rigorously by Chung and Lu, as the order of the graph approaches infinity, this model generates the scale-free graphs that are so prevalent in biology and social networks [10]. Geometry may be seen to emerge from this process by exploiting our construction of invariant stars.

Consider an invariant (not necessarily configured) star in $\mathbb{R}^{N}$ with internal vertex located at the origin and with d external vertices. Let $\mathbf{x} \in \mathbb{R}^{N}$ be the center of mass of the external vertices. Suppose that x ≠ 0. Let b > 0 denote the distance of the center of mass from the origin along the axis of the star.

Lemma 7.2 ([2]).

The addition of a new external vertex at any point other than − db along the axis of the star produces a new invariant star. Furthermore, if γ denotes the invariant of the original star and $x \in \mathbb{R}$ is the position along the axis of the star, then the new star invariant is given by

$$\displaystyle{ \widetilde{\gamma }= \frac{(d + 1)(x^{2} + db^{2}\gamma )} {(x + db)^{2}} \,. }$$

(7.29)

Proof.

Without loss of generality, we may suppose that the center of mass of the star lies along the y ^N-axis. In particular, if v ₁, …, v _d denote the external vertices, then

$$\displaystyle{ \sum _{\ell=1}^{d}\mathbf{v}_{\ell} = db\mathbf{e}_{ N}\,. }$$

We now add a new vertex at the point x e _N , for some $x \in \mathbb{R}$. Thus the new star matrix is given by

$$\displaystyle{ \left (\mathbf{v}_{1}\vert \cdots \vert \mathbf{v}_{d}\vert x\mathbf{e}_{N}\right )\,. }$$

Let A = (a _jk ) be an arbitrary orthogonal transformation of $\mathbb{R}^{N}$ and let $\phi: \mathbb{R}^{N} \rightarrow \mathbb{C}$ be the projection ϕ( y ¹, …, y ^N) = y ¹ + iy ². Set z _ℓ = ϕ ∘ A(v _ℓ ) for ℓ = 1, …, d and z _d+1 = ϕ ∘ A(x e _N ). Then

$$\displaystyle{ z_{\ell} =\sum _{ j=1}^{N}(a_{ 1j} + \mathrm{i}a_{2j})v_{\ell}^{j}\,,\qquad z_{ d+1} = x(a_{1N} + \mathrm{i}a_{2N})\,. }$$

Furthermore, ∑ _{ℓ = 1} ^d z _ℓ = db(a _1N + ia _2N ), so that

$$\displaystyle{ \sum _{\ell=1}^{d}z_{\ell}^{2} = \frac{\gamma } {d}\left (\sum _{\ell=1}^{d}z_{\ell}\right )^{2} = db^{2}\gamma (a_{ 1N} + \mathrm{i}a_{2N})^{2}\,, }$$

where γ is the invariant of the original star. We require that there is a real number $\widetilde{\gamma }$ such that

$$\displaystyle{ \widetilde{\gamma }\left (z_{d+1} +\sum _{ \ell=1}^{d}z_{\ell}\right )^{2} = (d + 1)\left (z_{ d+1}^{2} +\sum _{ \ell =1}^{d}z_{\ell}^{2}\right )\,. }$$

But this is uniquely given by (7.29). □

We note that as x approaches − db, then $\vert \widetilde{\gamma }\vert$ becomes arbitrary large. Indeed, when x = −db, then the Laplacian of ϕ vanishes at the internal vertex of the new star, so that $\widetilde{\gamma }$ is not well-defined in this case.

Suppose we are given a solution to (7.2) on a graph G = (V, E). For a given x ∈ V, let {y ₁, …, y _d } be its neighbors. Then by the lifting property described in Sect. 7.3, there is an invariant star K in $\mathbb{R}^{3}$ (or $\mathbb{R}^{2}$ if the degree of x is two) whose external vertices project to the values ϕ( y ₁), …, ϕ( y _d ) and whose internal vertex projects to ϕ(x). Introduce a new vertex y and form the graph $\widetilde{G} = (\widetilde{V },\widetilde{E})$ for which $\widetilde{V } = V \cup \{ y\}$ and $\widetilde{E} = E \cup \{\overline{xy}\}$. Define $\widetilde{\phi }:\widetilde{ V } \rightarrow \mathbb{C}$ by $\widetilde{\phi }(u) =\phi (u)$ for u ∈ V and $\widetilde{\phi }(\,y) = z$, where z is the projection of any point along the axis K. Then $\widetilde{\phi }$ satisfies (7.2) on $\widetilde{G}$ with γ modified according to the above lemma (note that any vertex of degree one always satisfies (7.2) with γ = 1). The choice of distance along the axis of the star may depend on some other parameter of the model, for example, so as to best uniformize γ.