Resistance Distances in Vertex-Face Graphs

1 Introduction

Suppose that G=(V(G), E(G)) is a connected edge-weighted graph (for convenience, we say a connected weighted graph) and the weight of each edge e=ij of G is denoted by w(e) or w_ij which is a positive real number. As a convention, when we say a graph, we mean a weighted graph with w(e)=1 for all edges. For any connected weighted graph G=(V(G), E(G)), we can view it as an electrical network with w(e) as the conductance of edge e [i.e. 1/w(e) is the resistance of e]. We denote the effective resistance of the network between u, v∈V(G) by r_uv(G). The effective resistance is a distance function (or metric) on graphs, which was first proved by Sharpe [1], and then rediscovered by Gvishiani and Gurvich [2], also by Klein and Randić [3]. In [3], Klein and Randić named this distance function as the “resistance distance” [3].

The resistance distance has attracted extensive attention due to their wide applications in physics, chemistry and others. Besides a number of well-known techniques developed by electrical engineers, such as, series and parallel principles, star-triangle transformation (or general star-mesh transformation) [4], [5], various formulae for computing resistance distances were derived, including algebraic formulae, probabilistic formulae, combinatorial formulae, sum rules, recursion formulae and so on, for more details, see [6], [7], [8], [9], [10], [11], [12], [13], [14] and references therein. Using the above techniques and formulae, resistance distances for many interesting (classes of) graphs have been computed, specially for some transformation graphs, such as, the line graph, the triangulation graph, the subdivision-vertex join, etc. [15], [16], [17], [18], [19], [20]. In this paper, we shall consider the resistance distances in vertex-face graphs.

Suppose that G is a triangulation graph embedded on an orientable surface Σ with genus g with vertex set V(G)={v₁, v₂,…,v_n}, edge set E(G)={e₁, e₂,…,e_m}, and face set F(G)={ϕ₁, ϕ₂,…,ϕ_f}. Let V(ϕ) denote the set of vertices on the boundary of face ϕ of G. Note that each face ϕ of G is a triangle. So |V(ϕ)|=3. Now we define a new graph K(G), which is called the vertex-face graph of G, from G as follows. The vertex set V(K(G)) of K(G) is the union of V(G) and F(G), i.e.

V(K(G))=V(G)∪F(G),

and the edge set E(K(G)) of K(G) is

E(G)∪{(u, ϕ)|u∈V(ϕ), ϕ∈F(G)}.

If G is the complete graph with four vertices which is embedded on the plane as illustrated in Figure 1a, then the corresponding vertex-face graph K(G) can be illustrated in Figure 1b. Obviously, K(G) is also a triangulation with n+f vertices embedded on Σ. So we generate a graph sequence K⁰(G)=G, K^k(G)=K(K^k−1(G)), k=1, 2,…. Let n_k=|V(K^k(G))|, and f_k=|F(K^kG))|, then by the Eulerian formula n–m+f=2−2g and 2m=3f, we have

Figure 1:

A plane triangulation graph and its corresponding vertex-face graph. (a) A plane triangulation G with vertex set V(G)={1, 2, 3, 4} and face set F(G)={ϕ₀₁, ϕ₀₂, ϕ₀₃, ϕ₀₄}. (b) The vertex-face graph K(G) of G.

nk=3kn+(3k−1)(2g−2), fk=2⋅3k(n+2g−2).

The vertex-face graph was introduced in [21], where the Laplacian spectrum, the normalised Laplacian spectrum, the number of spanning trees, the Kirchhoff index and the degree-Kirchhoff index of K(G) were considered. Here we concentrate on resistance distances in K(G). In the next section, using the well-known star-triangle transformation and resistance sum rules given in [8], we first obtain the explicit relations between resistance distances in K(G) and those in G. Then in Section 3, as applications, resistance distances in several networks are computed, including the modified Apollonian network K^k(K₃) and networks constructed from tetrahedron, octahedron and icosahedron.

2 Resistance Distances in Vertex-Face Graphs

For simplicity, in the following, for any weighted graph G, we always view the multiple edges (if they exist) as a single edge whose weight equals the sum of weights of the multiple edges. So we always assume that the underlying graph is simple and connected. For a weighted graph G=(V(G), E(G)) with n vertices, i, j∈V(G), we write i~j if e=ij∈E(G); i≁j otherwise. The weight of a vertex i is defined as w_i=∑_k∈N(i)w_ik, where N(i) denotes the set of vertices adjacent to i. Let R(G)=(r_ij(G)) denote the resistance matrix of G. In this section, we shall discuss the resistance distances and resistance matrix of vertex-face graphs. First we recall the well-known star-triangle transformation (see Fig. 2) [5], the weights satisfy:

Figure 2:

The star-triangle transformation (a) The star. (b) The triangle.

w12=w14w24w14+w24+w34, w23=w24w34w14+w24+w34,w13=w14w34w14+w24+w34.

Lemma 2.1 ([8]):LetGbea connected weighted graph of order n, Then

∑uv∈E(G)wuvruv=n−1,

wirij+∑k∈N(i)wik(rik−rjk)=2,

for any two distinct vertices v_i and v_j.

Theorem 2.2:Suppose thatGis a triangulation graph embedded on an orientable surface ∑ with vertex setV(G)={v₁, v₂,…,v_n} and face setF(G)={ϕ₁, ϕ₂,…,ϕ_f}. Then for the vertex-face graphK(G) ofGwithV(K(G))=V(G)∪F(G), we have

1. if v_i, v_j ∈V(G), then,

   rij(K(G))=35rij(G);

2. if v_i∈V(G), ϕ_j∈F(G), V(ϕ_j)={a, b, c}, then,

   rij(K(G))=13+15∑k∈V(ϕj)rik(G)−115rϕj(G),

   where rϕj(G)=rab(G)+rac(G)+rbc(G);

3. if ϕ_i, ϕ_j∈F(G), V(ϕ_i)={d, e, f}, V(ϕ_j)={a, b, c}, then,

   rij(K(G))=23+115(∑k∈V(ϕi), l∈V(ϕj)rkl(G)−rϕj(G)−rϕi(G)).

Proof. (i) Note that V(K(G))=V(G)∪F(G), every vertex in F(G) with its three neighbours in K(G) induces a star. Applying star-triangle transformation on these stars, we obtain a weighted graph G′ with the same underlying graph as G except that the weight is 53 for each edge (for an explanation example, see Fig. 3). So the result follows immediately.

(ii) For v_i∈V(G), ϕ_j∈F(G) with V(ϕ_j)={a, b, c} (see Fig. 4), first by Lemma 2.1,

Figure 3:

Star-triangle transformation be applied to the four stars with four face vertices as the centres. The new edge-weight is 1+13+13=53.

Figure 4:

v_i∈V(G), ϕ_j∈F(G).

{3rja(K(G))+∑k∈V(ϕj)(rjk(K(G))−rak(K(G)))=23rjb(K(G))+∑k∈V(ϕj)(rjk(K(G))−rbk(K(G)))=23rjc(K(G))+∑k∈V(ϕj)(rjk(K(G))−rck(K(G)))=2.

That is

{4rja(K(G))+rjb(K(G))+rjc(K(G)))=2+rab(K(G)) +rac(K(G))rja(K(G))+4rjb(K(G))+rjc(K(G)))=2+rab(K(G)) +rbc(K(G))rja(K(G))+rjb(K(G))+4rjc(K(G)))=2+rbc(K(G)) +rac(K(G)).

Solving this equation system, we have

{rja(K(G))=13+19(2rab(K(G))+2rac(K(G))−rbc(K(G)))rjb(K(G))=13+19(2rab(K(G))+2rbc(K(G))−rac(K(G)))rjc(K(G))=13+19(2rac(K(G))+2rbc(K(G))−rab(K(G))).

So

Now for v_i∈V(G), ϕ_j∈F(G), using Lemma 2.1 again, we have

3rji(K(G))+rja(K(G))+rjb(K(G))+rjc(K(G))−ria(K(G)) −rib(K(G))−ric(K(G))=2.

Then by (1) and (i), we obtain

rij(K(G)) =13+13(ria(K(G))+rib(K(G))+ric(K(G))) −19(rab(K(G))+rbc(K(G))+rac(K(G))) =13+15(ria(G)+rib(G)+ric(G))−115(rab(G) +rbc(G)+rac(G)).

(iii) If ϕ_i, ϕ_j∈F(G) with V(ϕ_i)={d, e, f}, V(ϕ_j)={a, b, c} (see Fig. 5). First by (ii), we have

Figure 5:

ϕ_i, ϕ_j∈F(G).

By Lemma 2.1 again,

3rji(K(G))+rja(K(G))+rjb(K(G))+rjc(K(G))−ria(K(G)) −rib(K(G))−ric(K(G))=2.

Substituting (1) and (2) into this equation, we obtain

rij(K(G))=23+115(∑k∈V(ϕi), l∈V(ϕj)rkl(G)−rϕj(G)−rϕi(G)).

Now in order to calculate resistance distances by the computer, we write the relations in Theorem 2.2 in matrix form. Suppose that G is a triangulation embedded on an orientable surface Σ with vertex set V(G)={v₁, v₂,…,v_n} and face set F(G)={ϕ₁, ϕ₂,…,ϕ_f}. Define the vertex-face incident matrix as M(G)={m_ij}_n×_f, where m_ij=1 if vertex v_i∈V(ϕ_j) and m_ij=0 otherwise. For the triangulation G illustrated in Figure 1a, V(G)={1, 2, 3, 4} and F(G)={ϕ₀₁, ϕ₀₂, ϕ₀₃, ϕ₀₄}, the vertex-face incident matrix M(G) is

(1011110101111110).

Let J_n×f , I_f×f denote the n×f all-ones matrix and the identity matrix of order f, respectively. Write diag(a₁, a₂,…,a_n) for the diagonal matrix with diagonal elements a₁, a₂,…,a_n. Then the relations obtained in Theorem 2.2 can be expressed in the block-matrix form as follows:

R(K(G))=(R1R2R2TR3),

where

R1=35R(G),

R2=13Jn×f+15R(G)M(G)−115Jn×fdiag(Rϕ1(G),…,Rϕf(G)),

R3=23(Jf×f−If×f)+115M(G)ΤR(G)M(G)−115(Jf×fdiag(Rϕ1(G),…,Rϕf(G))+diag(Rϕ1(G),…,Rϕf(G))Jf×f).

So if we know R(G), we can get R(K^k(G)) for any k≥1 by iterating the process. In the next section, we shall compute resistance distances in some networks embedded on the plane.

3 Applications

Nowadays, research on the resistor network has expanded into the basic model in the field of science. Looking for the exact solution of resistance between any two nodes in a resistor network in the finite case is important but difficult. The resistance between two nodes in some resistor networks has been studied extensively by mathematicians and physicists, for example, lattice network, fan network, cobweb network, etc. [22], [23], [24]. In this section, we shall give some exact expressions for resistances in several networks constructed by iterating vertex-face operation.

Suppose that G is a triangulation embedded on the plane with vertex set V(G)={v₁, v₂,…,v_n} and face set F(G)={ϕ₁, ϕ₂,…,ϕ_f}. Then for the vertex-face graph sequence K⁰(G)=G, K^k(G)=K(K^k−1(G)), k=1, 2,…. Since the genus of the plane is 0, we have

nk= |V(Kk(G))| =(n−2)3k+2  and fk=|F(Kk(G))|=2(n−2)3k.

By the definition of vertex-face graph, we can write

V(Kk(G))=V(G)∪i=0k−1F(Ki(G)).

For simplicity and clearness, in the following, we always write as

V(G)={1, 2,…,n}, F(Kk(G))={ϕk1, ϕk2,…,ϕkfk}.

Now, we use the results obtained in the above section to compute resistance distances in the network K^k(G) when G is the complete graph K₃, tetrahedron, octahedron and icosahedron embedded on the plane, respectively.

Example 3.1: The Complete Graph K₃

The graphs of K^k(K₃) for k=0, 1, 2 are shown in Figure 6.

Figure 6:

A modified Apollonian network K^k(K₃) produced at iterations k=0, 1, 2.

As

R(K3)=23(011101110),

by Theorem 2.2, we can easily obtain the resistance matrices

and

Remark: Note that K^k(K₃) is called the modified Apollonian network in [25] as its construction process is only a little different from that of the Apollonian network (see Fig. 7 for an illustration) [26], [27], [28]. In [25], the author computed the number of spanning trees of K^k(K₃). Here we give some results on resistance distances for the modified Apollonian network.

Figure 7:

An Apollonian network A^(k) produced at iterations k=0, 1, 2, 3.

Theorem 3.1:For the modified Apollonian networkK^k(K₃)(k≥2) with vertex setV(Kk(K3))=V(K3)∪i=0k−1F(Ki(K3)),wehave

1. if i, j∈V(K₃), then

   rij(Kk(K3))=(35)k23;

2. if i∈V(K₃), j∈F(K₃), then

   rij(Kk(K3))=(35)k−1715;

3. if i, j∈F(K₃), then

   rij(Kk(K3))=(35)k−123;

4. if i∈V(K₃), j∈F(K(K₃)), then

   rij(Kk(K3))={(35)k−294225, i~j,(35)k−2112225, i≁j;

5. if i∈F(K₃), j∈F(K(K₃)), then

   rij(Kk(K3))={(35)k−297225, i~j,(35)k−2127225, i≁j;

6. if i, j∈F(K(K₃)), then

   rij(Kk(K3)) ={(35)k−2156225, V(ϕi)∩V(ϕj)={k, l}, k∈V(K3),        l∈F(K3),(35)k−2160225, V(ϕi)∩V(ϕj)={k, l}, k, l∈V(K3),(35)k−2166225, V(ϕi)∩V(ϕj)={k}, k∈F(K3).

Example 3.2: The Tetrahedron Graph

For the tetrahedron G illustrated in Figure 1a, the corresponding vertex-face graphs K(G) is illustrated in Figure 1b. From

R(G)=12(0111101111011110),

we obtain

Theorem 3.2:For the networkK^k(G)(k≥1) with vertex setV(Kk(G))=V(G)∪i=0k−1F(Ki(G)),whereGis the tetrahedron graph. Then we have

1. if i, j∈V(G), then

   rij(Kk(G))=(35)k12;

2. if i∈V(G), j∈V(F(G)), then

   rij(Kk(G))={(35)k−11330, i~j,(35)k−1815, i≁j;

3. if i, j∈V(F(G)), then

   rij(Kk(G))=(35)k−1710.

Example 3.3: The Octahedron Graph

The octahedron graph G is illustrated in Figure 8a, and the corresponding vertex-face graph K(G) is illustrated in Figure 8b. The resistance distances had been obtained in [7] and [9], that is

Figure 8:

The octahedron and its corresponding vertex-face graph. (a) The octahedron; (b) the vertex-face graph of the octahedron.

R(G)=112(055655505556550565655055556505565550).

So we can obtain that

Theorem 3.3:For the networkK^k(G)(k≥1) with vertex setV(Kk(G))=V(G)∪i=0k−1F(Ki(G)),whereGis the octahedron graph. We have

1. if i, j∈V(G), then

   rij(Kk(G))={(35)k512, i~j,(35)k12, i≁j;

2. if i∈V(G), j∈V(F(G)), then

   rij(Kk(G))={(35)k−1512, i~j,(35)k−13160, i≁j;

3. if i, j∈V(F(G)), then

   rij(Kk(G))={(35)k−1710, |V(ϕi)∩V(ϕj)| =2,(35)k−11115, |V(ϕi)∩V(ϕj)| =1,(35)k−12330, |V(ϕi)∩V(ϕj)| =0.

Example 3.4: The Icosahedron Graph

The Icosahedron graph G is illustrated in Figure 9. Let d_ij denote the shortest-path distance of i and j in G, then it has been obtained that [7], [9]

Figure 9:

The icosahedron graph.

rij(G)={1130, dij=1,715, dij=2,12, dij=3.

Theorem 3.4:For the networkK^k(G)(k≥1) with vertex setV(Kk(G))=V(G)∪i=0k−1F(Ki(G)),whereGis the Icosahedron graph. We have

1. if i, j∈V(G), then

   rij(Kk(G))={(35)k1130, dij=1,(35)k715, dij=2,(35)k12, dij=3;

2. if i∈V(G), j∈V(F(G)), V(φ_j)={a, b, c}, then

   rij(Kk(G))={(35)k−161150, Γ1={0, 1, 1},(35)k−112, Γ1={1, 1, 2},(35)k−11325, Γ1={1, 2, 2},(35)k−14175, Γ1={2, 2, 3},

   where the multiple set Γ₁={d_ia, d_ib, d_ic};

3. if i, j∈V(F(G)), V(ϕ_i)={d, e, f}, V(ϕ_j)={a, b, c}, then

   rij(Kk(G))={(35)k−1157225, Γ2={0(), 1(), 2},(35)k−1331450, Γ2={0, 1(5), 2(3)},(35)k−1176225, Γ2={1(), 2(), 3},(35)k−1359450, Γ2={1, 2(), 3()},(35)k−1121150, Γ2={2(), 3()},

   where the multiple set Γ₂={d_kl|k∈V(ϕ_i), l∈V(ϕ_j)}.