Elsevier

Discrete Applied Mathematics

Volume 284, 30 September 2020, Pages 341-352
Discrete Applied Mathematics

Spanning 2-forests and resistance distance in 2-connected graphs

https://doi.org/10.1016/j.dam.2020.03.061Get rights and content

Abstract

A spanning 2-forest separating vertices u and v of an undirected connected graph is a spanning forest with 2 components such that u and v are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices u and v in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating u and v divided by the number of spanning trees in the graph. There are also well-known matrix theoretic methods for calculating resistance distance, but the way in which the structure of the underlying graph determines resistance distance via these methods is not well understood.

For any connected graph G with a 2-separator separating vertices u and v, we show that the number of spanning trees and spanning 2-forests separating u and v can be expressed in terms of these same quantities for the smaller separated graphs, which makes computation significantly more tractable. An important special case is the preservation of the number of spanning 2-forests if u and v are in the same smaller graph. In this paper we demonstrate that this method of calculating resistance distance is more suitable for certain structured families of graphs than the more standard methods. We apply our results to count the number of spanning 2-forests and calculate the resistance distance in a family of Sierpinski triangles and in the family of linear 2-trees with a single bend.

Introduction

Resistance distance in graphs has played a prominent role not only in circuit theory and chemistry [1], [5], [10], [12], [15], but also in combinatorial matrix theory [2], [17] and spectral graph theory [1], [5], [9], [14]. Many of the methods for calculating resistance distance, e.g., those making use of the Laplacian matrix are O(n3) where n is the number of vertices of the graph. Furthermore, the relationship between these resistance distances and the structure of the underlying graph is not well understood except in special cases. An under-utilized method of calculating the resistance distance between two vertices u and v in a graph G is by determining the number of spanning 2-forests separating u and v in G and the number of spanning trees of G (see Definition 1 and Theorem 2). Thus, if the number of spanning trees is known, calculating the number of spanning 2-forests and resistance distance are equivalent problems. This work presents new reduction formulas for determining these quantities for 2-connected graphs. We apply these results to a new family of linear 2-trees generalizing the work of [4]. We begin with the following notation and definitions.

Let G be an undirected graph in which multiple edges are allowed but loops are not. Let V(G) denote the vertex set of G and unless otherwise specified V(G)={1,2,,n}. Finally, let T(G) denote the number of spanning trees of G.

Definition 1

Given any two vertices u and v of G, a spanning 2-forest separating u and v is a spanning forest with two components such that u and v are in distinct components. The number of such forests is denoted by FG(u,v). In addition, we occasionally consider spanning 2-forests separating a vertex u from a pair of vertices v and w. We denote the number of these by FG(u,{v,w}).

It follows from the matrix tree theorem [6, p. 5] that for any j{1,2,,n}, T(G)=detLG(j) where LG is the combinatorial Laplacian matrix of G, and LG(j) is the matrix obtained from LG by deleting the jth row and column. The following identity (see [7] and Th. 4 of [1]) is a relative of the matrix tree theorem: FG(u,v)=detLG(u,v),where LG(u,v) is the matrix obtained from LG by deleting rows u,v and columns u,v.

If G has a cut-vertex w and G=G1G2 with G1 and G2 connected and V(G1)V(G2)={w}, then it is evident that T(G)=T(G1)T(G2)FG(u,v)=FG1(u,v)T(G2)foru,vV(G1)FG(u,v)=FG1(u,w)T(G2)+T(G1)FG2(w,v)foruV(G1)andvV(G2).

It is natural to ask if reduction formulas such as these can be found for graphs with no cut vertex, and the answer is in the affirmative if the graph has a cut-set of size 2. For any graph G with a 2-separator {i,j} and associated decomposition G=G1G2, one can express T(G) in terms of T(Gk) and FGk(i,j),k=1,2, and, furthermore, for any pair of vertices u,vV(G) one can express FG(u,v) in terms of T(Gk),FGk(x,y), and FGij(x,ij) for k{1,2}, x{u,v} and y{i,j}. Here Gij denotes the graph obtained by identifying vertices i and j (see Definition 7). This is Theorem 13 and is one of the main results of the next section. This reduction is particularly effective if the sizes of G1 and G2 are comparable, and if there are multiple 2-separators.

As previously mentioned we also consider the important and closely related concept of resistance distance or effective resistance. Consider G as an electric circuit with unit resistance on each edge, and suppose one unit of current flows into vertex u and one unit of current flows out of vertex v. Then the resistance distance rG(u,v) between vertices u and v is the “effective” resistance between u and v. Alternatively, one can give a mathematical formulation rG(u,v)=(euev)TLG(euev),where denotes the Moore–Penrose inverse. The following theorem [1, Th. 4 and (5)] gives the relationship between the resistance distance between u and v and the number of spanning 2-forests separating u and v.

Theorem 2

Given a graph G, the resistance distance between vertices u and v is given by rG(u,v)=FG(u,v)T(G).

Returning to the case where G has a cut-vertex w as described above, we divide (2) by T(G1)T(G2) and applying Theorem 2 we obtain rG(u,v)=rG1(u,v), a much shorter proof than the one given of the same result, Theorem 2.5 (Cut Vertex Theorem) in [4]. Dividing (3) by T(G1)T(G2) we see that if w is a cut vertex of G and u and v lie in distinct components of Gw, then rG(u,v)=rG1(u,w)+rG2(v,w).

Aside from (4), there seem to be few applications of Theorem 2 to the calculation of resistance distance. One significant example is the proof of the second statement of Theorem 7 in [3]. Our reduction formulas open the possibility of finding closed forms for resistance distances in many additional graphs. We illustrate this for the Sierpinski triangle and the family of linear 2-trees with a single bend in Section 3 (see Fig. 4, Fig. 6).

Definition 3

A linear 2-tree (or 2-path) on n vertices is a graph G satisfying the following 4 properties.

  • G has 2n3 edges.

  • K4 is not a subgraph of G.

  • G is chordal (every induced cycle is a triangle).

  • G has two degree two vertices.

Alternatively, a linear 2-tree is a graph G that is constructed inductively by starting with a triangle and connecting each new vertex to the vertices of an existing edge that includes a vertex of degree 2.

Definition 4 Straight Linear 2-tree

A straight linear 2-tree is a graph Gn with n vertices with adjacency matrix that is symmetric, banded, with the first and second subdiagonals equal to one, the first and second superdiagonals equal to one, and all other entries equal to zero. See Fig. 1.

In [4] the authors obtained an explicit formula for the resistance distance between any two vertices in a straight linear 2-tree on n vertices, and verified that the number of spanning trees of a straight linear 2-tree is F2n2, where Fk is the kth Fibonacci number. Consequently, the number of spanning 2-forests separating two vertices can be found immediately from Theorem 2.

A linear 2-tree with a single bend can be obtained from two straight linear 2-trees. This fact and Theorem 13 are applied in Section 3 to obtain an explicit formula for all resistance distances (all separating 2-forests) in the family of linear 2-trees with a single bend. If u and v are the end vertices of this “bent” linear 2-tree on n vertices, the number of spanning 2-forests separating u and v is less than the number in the straight linear 2-tree by the product of four Fibonacci numbers. (See Corollary 25)

Section snippets

2-separations

Definition 5

A 2-separation of a graph G is a pair of subgraphs G1,G2 such that

  • V(G)=V(G1)V(G2),

  • |V(G1)V(G2)|=2,

  • E(G)=E(G1)E(G2), and

  • E(G1)E(G2)=.

The pair of vertices, V(G1)V(G2), is called a 2-separator of G.

Throughout this section, we will let G denote a graph with a 2-separation, G1,G2 will denote the two graphs of the separation, and we will let {i,j}=V(G1)V(G2). Note that the graph resulting from the deletion of vertices i and j from G is a disconnected graph.

Theorem 6

Let G be a graph with a 2-separation as

Resistance distance and spanning 2-forests in the sierpinski triangle

We first demonstrate the utility of Theorem 18 by applying it to give a very short argument that the resistance distance rn(a,b) between two vertices of degree 2 in the stage n Sierpinski triangle Sn is 2353n. This is one of the results obtained in [11] by means of series, parallel, and Delta–Wye transformations. The first three stages are shown in Fig. 4.

It is straightforward to verify that r0(a,b)=23 for any two vertices a,b, of S0. Now, we assume that rn(a,b)=x, and use the separation shown

Conclusion

In this paper we have given a non-trivial generalization of well-known and elementary formulas for calculating the number of spanning trees of a graph G and the number of separating 2-forests separating 2 vertices in the case that G contains a cut-vertex, to the case in which G contains a 2-separator. We have applied these formulas to the Sierpinski triangle and to a family of linear 2-trees with a single bend. For the Sierpinski triangle we observed that for any of the degree 2 vertices a and b

CRediT authorship contribution statement

Wayne Barrett: Conceptualization, Investigation, Writing - original draft, Project administration, Validation. Emily J. Evans: Conceptualization, Investigation, Writing - original draft, Project administration, Validation, Software, Visualization. Amanda E. Francis: Conceptualization, Investigation, Writing - original draft, Project administration, Validation, Software, Visualization. Mark Kempton: Conceptualization, Investigation, Writing - original draft, Project administration, Validation.

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Supported by the Defense Threat Reduction Agency, USA – Grant Number HDTRA1-15-1-0049.

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