Spanning 2-forests and resistance distance in 2-connected graphs
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 where 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 and in a graph is by determining the number of spanning 2-forests separating and in and the number of spanning trees of (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 be an undirected graph in which multiple edges are allowed but loops are not. Let denote the vertex set of and unless otherwise specified . Finally, let denote the number of spanning trees of .
Definition 1 Given any two vertices and of , a spanning 2-forest separating and is a spanning forest with two components such that and are in distinct components. The number of such forests is denoted by . In addition, we occasionally consider spanning 2-forests separating a vertex from a pair of vertices and . We denote the number of these by .
It follows from the matrix tree theorem [6, p. 5] that for any , where is the combinatorial Laplacian matrix of , and is the matrix obtained from by deleting the th row and column. The following identity (see [7] and Th. 4 of [1]) is a relative of the matrix tree theorem: where is the matrix obtained from by deleting rows and columns .
If has a cut-vertex and with and connected and , then it is evident that
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 with a 2-separator and associated decomposition , one can express in terms of and , and, furthermore, for any pair of vertices one can express in terms of , and for , and . Here denotes the graph obtained by identifying vertices and (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 and 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 as an electric circuit with unit resistance on each edge, and suppose one unit of current flows into vertex and one unit of current flows out of vertex . Then the resistance distance between vertices and is the “effective” resistance between and . Alternatively, one can give a mathematical formulation where denotes the Moore–Penrose inverse. The following theorem [1, Th. 4 and (5)] gives the relationship between the resistance distance between and and the number of spanning 2-forests separating and .
Theorem 2 Given a graph , the resistance distance between vertices and is given by
Returning to the case where has a cut-vertex as described above, we divide (2) by and applying Theorem 2 we obtain , a much shorter proof than the one given of the same result, Theorem 2.5 (Cut Vertex Theorem) in [4]. Dividing (3) by we see that if is a cut vertex of and and lie in distinct components of , then
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 vertices is a graph satisfying the following 4 properties. has edges. is not a subgraph of . is chordal (every induced cycle is a triangle). has two degree two vertices.
Alternatively, a linear 2-tree is a graph 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 with 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 vertices, and verified that the number of spanning trees of a straight linear 2-tree is , where is the th 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 and are the end vertices of this “bent” linear 2-tree on vertices, the number of spanning 2-forests separating and 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 is a pair of subgraphs such that , , , and .
The pair of vertices, , is called a 2-separator of .
Throughout this section, we will let denote a graph with a 2-separation, will denote the two graphs of the separation, and we will let . Note that the graph resulting from the deletion of vertices and from is a disconnected graph.
Theorem 6 Let 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 between two vertices of degree 2 in the stage Sierpinski triangle is . 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 for any two vertices , of . Now, we assume that , 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 and the number of separating 2-forests separating 2 vertices in the case that contains a cut-vertex, to the case in which 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 and
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.