Abstract
The resistance distance between any two vertices of a connected graph is defined as the net effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. In this article, using electric network approach and combinatorial approach, we derive exact expression for resistance distances between any two vertices of polyacene graphs.
1 Introduction
Let be a connected graph. It is interesting to consider distance functions on G. The most natural and best-known distance function is the shortest path distance. For any two vertices , the shortest path distance between i and j, denoted by , is defined as the length of a shortest path connecting i and j. Two decays ago, another novel distance function, named resistance distance, was identified by Klein and Randić [1]. The concept of resistance distance originates from electrical circuit theory. If we view G as an electrical network N by replacing each edge of G with a unit resistor, then the resistance distance [1] between i and j, denoted by , is defined as the net effective resistance between the corresponding nodes in the electrical network N. In contrast to the shortest path distance, the resistance distance has a notable feature that if i and j are connected by more than one path, then they are closer than they are connected by the only shortest path. So it is suggested that resistance distance is more appropriate to deal with wave-like motion in the network, like the communication in chemical molecules. In addition, it turns out that the resistance distance has some pure mathematical interpretations, which could be expressed in terms of the generalized inverse of the Laplacian matrix [1], the number of spanning trees and spanning bi-trees [2], and random walks on graphs [3, 4].
Besides being an intrinsic graph metric and an important component of electrical circuit theory, resistance distance also turns out to have important applications in chemistry. For this reason, resistance distance has been widely studied in the mathematical, chemical, and physical literature. In the study of resistance distance, the main focus is placed on the problem of computation of resistance distance. This problem has been a classical problem in electrical network theory studied by numerous researchers for a long time. Besides, it is also relevant to a wide range of problems ranging from random walks, the theory of harmonic functions, to lattice Green’s functions. Consequently, this problem has attracted much attention, and many researchers have devoted themselves to it. Up to now, resistance distances have been computed for many interesting (classes of) graphs, with emphasis being placed on some highly concerned electrical networks and chemical interesting graphs. For example, resistance distances have been computed for Platonic solids [5], and for some fullerene graphs including buckminsterfullerene [6], circulant graphs [7], distance-regular graphs [8, 9], pseudo–distance-regular graphs [10], wheels and fans [11], Cayley graphs over finite abelian groups [12], complete graph minus N edges [13], resistor network embedded on a globe [14], Möbius ladder [15], cobweb network [16], complete n-partite graphs [17], resistor network [18], ladder graph [19], n-step network [20], Cayley graphs on symmetric groups [21], Apollonian network [22], Sierpinski Gasket Network [23], generalized decorated square and simple cubic network lattices [24], self-similar -flower networks [25], almost complete bipartite graphs [26], straight linear 2-trees [27], and path networks [28].
It is interesting to note that a good deal of attention has been paid on resistance distances in plane networks, such as Platonic solids, fullerene graphs, wheels, fans, ladder graphs, Apollonian network, Sierpinski Gasket Network, resistor network, and straight linear 2-tree. Motivated by this fact, we are devoted to considering other interesting plane networks. In this article, we take the linear polyacene graphs into consideration. It is well known that the linear polyacene graphs are graph representations of an important class of benzenoid hydrocarbons, and it is an interesting class of plane hexagonal networks. We use to denote the linear polyacene graph with benzenoid rings (i.e., hexagons), as shown in Figure 1. Using electrical network approach and resistance distance local rules, we derive exact expression for resistance distances between any two vertices of .
FIGURE 1
2 Resistance Distances in Linear Polyacene Graphs
Let be the linear polyacene graph with benzenoid rings. Obviously, has vertices and edges. For convenience, we label the vertices in as in Figure 1. We partite the vertex set of into two classes: and . To compute resistance distances between any two vertices of , we take two steps. In the first step, we compute resistance distances between vertices in . To this end, we first view as a weighted ladder graph by simply replacing all the paths and by edges of resistance 2. Then, by making use of the electric network approach as inspired in [19], we obtain resistance distances between vertices in . Next, for the second step, using the results obtained in the first step together with resistance distance local rules, we derive expressions for resistance distances between the remaining pairs of vertices.
Before stating the main result, we introduce the elegant resistance distance local rules, which will be frequently used later. For any vertex , we use to denote the set of neighbors of a. Then, we have the following sum rules for resistance distances.
Lemma 2.1 [29]. Let be a connected graph with vertices. Then,where denotes the degree of the vertex a.
2) For any three different vertices ,Now, we are ready for the main theorem. For simplicity, we let , and define and as follows:Then, the main result is given in the following.
Theorem 2.2. The resistance distances between any two vertices in the linear polyacene graph can be computed as follows.Proof. We divide the proof into two steps.
Step 1. Computation of resistance distances between any two vertices in .
To compute resistance distances between vertices in , we view as a weighted ladder graph by simply replacing all the paths and by edges of resistance 2, see Figure 2 (left). Clearly, holds for all .
FIGURE 2
First, we compute resistance distances between the end vertices , , , and . let , , and . Clearly, can be obtained from by adding two vertices and , and the three edges with end vertices , , and , as shown in Figure 2 (right). Hence, according to rules for series and parallel circuits, could be expressed in term of aswith initial condition . Solving the recurrence relation by Mathematica [30], we obtainSpecially, we have , , , and . It is easily checked that can also be expressed asWe proceed to use to find explicit formulas for and . To this end, we make circuit reduction to the subgraph of with respect to , , and , where . Precisely speaking, we reduce to a Y-shaped graph which has outer vertices , , and . We use A, B, and C to denote the effective resistances between end vertices of those edges of the Y-shaped graph. Then, we have , , and . Solving these equations, we getOn the other hand, by parallel and series connection rules, we have and . So, it follows thatwith initial conditions and . Eq. 2.10 minus Eq. 2.11 yieldsSet . It follows thatThus, we haveSince , using Eq. 2.9 and doing some algebraic calculations, we getThis could also be rewritten as , for all . Now, we come back to solve and . By using , Eqs 2.8–2.14 and doing some algebra, Eq. 2.11 becomesSolving the recursion relation, we getNow, by Eqs 2.14–2.16, together with the relation , we getNext, we proceed to compute , , and , where . To achieve our goal, we consider as the union of three graphs: the upper part of and , the lower part of and , and the middle part consisting of , , , and , as shown in Figure 3. Note that the upper and the lower graphs are corresponding to the graphs and , respectively. We make circuit reductions as illustrated in Figure 3. First, make the circuit reduction of the upper part with respect to , , and to obtain a Y-shaped graph, and assume that resistances along its edges are M, N, and K. Then, reduce the lower part of and to be edge with resistance . We could find thatNote thatSolving M, N, and K, we obtainThen, applying parallel and series connection rules to the reduced circuit in Figure 3, we obtainSubstituting Eqs 2.8–2.20 into Eq. 2.21, we haveFinally, we compute and (). To this end, we consider as the union of two graphs: the upper part and the lower part with respect to and , as illustrated in Figure 4. Note the lower part is the graph , and the upper part is the graph . Next, we make circuit reduction to so that it is reduced to an edge with resistance . Then, we reduce to a Y-shaped graph with end vertices , , and , and resistances D, E, and F along its edges. These reductions are illustrated in Figure 4. Then, we haveIt follows thatOn the other hand, by the series and parallel connection rules, we haveBy Eqs. (2.8), Eqs 2.22–2.25, and doing some algebra using Mathematica [30], we obtainIt is easily verified that Eq. 2.27 is valid for .
FIGURE 3
FIGURE 4
Step 2. Computation of resistance distances between and between and .
First, we compute and . Applying Lemma 2.1 to pairs of vertices and , we obtainMultiplying Eq. 2.28 by 3 and then minus Eq. 2.29, we getThen, substituting the value of as obtained in Step 1 into Eq. 2.30, we could obtainSubstituting Eq. 2.31 into Eq. 2.30, we haveIn the same way, we could obtain thatSecond, we calculate the resistance distance between and . Again, applying Lemma 2.1 to , we obtainBy Eqs 2.32, 2.33, it follows thatFor the sake of simplicity, we defineThen, Eq. 2.35 can be rewritten asOn the other hand, by Eq. 2.26, we haveSubstituting Eqs. 2.37, 2.38 into Eq. 2.34, we draw the conclusion thatThird, we calculate the resistance distance between and . Apply Lemma 2.1 to to obtainBy Eq. 2.37, we haveFor simplicity, we defineOn the other hand, by Eq. 2.27, we haveSubstituting Eqs. 2.41–2.43 into Eq. 2.40, we getFourth, we calculate the resistance distance between and . Applying Lemma 2.1 to , we haveAs , , , and have been given by Eq. 2.39, simple calculation leads toFifth and finally, we calculate the resistance between and . Applying Lemma 2.1 to , we haveNote by the symmetry of that we have and . Using the results obtained in Eqs. 2.39–2.44, simple algebraic calculation yields
3 Conclusion
The computation of resistance distances is a classical problem in electrical circuit theory, which has attracted much attention. It is of special interest to investigate resistance distances in plane networks. Along this line, we have considered the linear polyacene network, with exact expression for resistance distances in this network being given. It is a primary attempt for the computation of resistance distances in plane hexagonal lattice. Resistance distances in more and more plane hexagonal lattices are greatly anticipated.
Funding
This research was funded by the National Natural Science Foundation of China through grant number 116711347, and project ZR2019YQ02 by Shandong Provincial Natural Science Foundation.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Acknowledgments
We would like to thank the anonymous reviewers for their useful comments.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
hexagonal lattice, local rules, polyacene graph, resistance distance, circuit reduction
Citation
Wang D and Yang Y (2021) Resistance Distances in Linear Polyacene Graphs. Front. Phys. 8:600960. doi: 10.3389/fphy.2020.600960
Received
31 August 2020
Accepted
24 November 2020
Published
12 January 2021
Volume
8 - 2020
Edited by
Andre P. Vieira, University of São Paulo, Brazil
Reviewed by
Zhibin Du, South China Normal University, China
Mohammad Reza Farahani, Iran University of Science and Technology, Iran
Updates
Copyright
© 2021 Wang and Yang.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Yujun Yang, yangyj@yahoo.com
This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics
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