Abstract
Let G be a connected graph with vertex set V(G). The resistance distance between any two vertices u, v ∈ V(G) is the net effective resistance between them in the electric network constructed from G by replacing each edge with a unit resistor. Let S ⊂ V(G) be a set of vertices such that all the vertices in S have the same neighborhood in G − S, and let G[S] be the subgraph induced by S. In this note, by the {1}-inverse of the Laplacian matrix of G, formula for resistance distances between vertices in S is obtained. It turns out that resistance distances between vertices in S could be given in terms of elements in the inverse matrix of an auxiliary matrix of the Laplacian matrix of G[S], which derives the reduction principle obtained in [J. Phys. A: Math. Theor. 41 (2008) 445203] by algebraic method.
1 Introduction
The novel concept of resistance distance was introduced by Klein and Randić [8] in 1993. For a connected graph G with vertex set V(G) = {1, 2, …, n}, the resistance distance between u, v ∈ V(G), denoted by ΩG (u, v), is defined to be the effective resistance between u and v in the corresponding electric network obtained from G by replacing each edge with a unit resistor. Since resistance distance is an intrinsic graph metric and an important component of circuit theory, with potential applications in chemistry, it has been extensively studied in mathematics, physics, and chemistry. For more information, we refer the readers to recent papers [2, 4, 6, 7, 10, 11, 15] and references therein.
Let G be a connected graph of order n. For any set of vertices U ⊂ V(G), we use G [U] to denote the subgraph induced by U, and G − U to denote the subgraph obtained from G by removing all the vertices in U as well as all the edges incident to vertices of U. The adjacency matrixAG of G is an n × n matrix such that the (i, j)-th element of AG is equal to 1 if vertices i and j are adjacent and 0 otherwise. The Laplacian matrix of G is LG = DG − AG, where DG is the diagonal matrix of vertex degrees of G. Clearly, LG is real symmetric and singular.
Let M be an n × m real matrix. An m × n real matrix X is called a {1}-inverse of M and denoted by M(1), if X satisfies the following equation:
If M is singular, then it has infinite {1}-inverses. It is well known that resistance distances in a connected graph G can be obtained from any {1}-inverse of LG (see [1]). So far, there are many well-established results on this inverse. For example, in 2014, Bu et al [4] obtained the {1}-inverse of the Laplacian matrix for a class of connected graphs, and investigated resistance distances in subdivision-vertex join and subdivision-edge join of graphs. Then in 2015, an exact expression for the {1}-inverse of the Laplacian matrix of connected graphs was obtained by Sun et al. [13]. After that, Liu et al. [9] obtained the {1}-inverses for the Laplacian matrix of subdivision-vertex and subdivision-edge coronae networks. Recently, Cao et al. [5] also characterised the {1}-inverses for the Laplacian of corona and neighborhood corona networks. Sardar et al. [12] determined resistance distances of some classes of rooted product graphs via the Laplacian {1}-inverses method.
In this paper, some results on the {1}-inverses for Laplacian matrices of graphs with given special properties are established. As an application, for any given vertex set S ⊂ V(G) such that all the vertices in S have the same neighborhood N in G − S, explicit formula for resistance distances between vertices in S is obtained. It turns out that resistance distances between vertices in S could be given in terms of elements in the inverse matrix of an auxiliary matrix of the Laplacian matrix of G[S], which derives the reduction principle obtained in [J. Phys. A: Math. Theor. 41 (2008) 445203] by algebraic method.
2 Preliminary Results
In this section, we present some preliminary results. We first introduce the concept of group inverse and Moore-Penrose inverse of a matrix.
Definition 2.1For a square matrix X, the group inverse of X, denoted by X#, is the unique matrix H that satisfies matrix equations:
Definition 2.2Let M be an n × m matrix. An m × n matrix X is called the Moore-Penrose inverse of M, if X satisfies the following conditions:where XH represents the conjugate transpose of the matrix X.If M is real symmetric, then there exists a unique M# and M# is the symmetric {1}-inverse of M. In particular, M# is equal to the Moore-Penrose inverse of M because M is symmetric [3].Let (M)ij denote the (i, j)-entry of M. It is well known that resistance distances in a connected graph G can be obtained from any {1}-inverse of LG according to the following lemma.
Lemma 2.3[3]LetGbe a connected graph. Then for verticesiandj,Let 0 and e be all-zero and all-one column vectors, respectively. Let Jn×m be the n × m all-one matrix. The following result is due to Sun et al. [13] which characterizes the {1}-inverse of the Laplacian matrix.
Lemma 2.4[13]Letbe the Laplacian matrix of a connected graph. IfL1is nonsingular, thenis a symmetric {1}-inverse ofLG, where.In particular, if each column vector of is − e or 0, then X can be further simplified. For convenience, in the rest of this section (see Lemmas 2.5, 2.6, 2.7), we always assume that , with the property that L1 is nonsingular, and each column vector of is − e or 0.
Lemma 2.5[4]LetLGbe defined as above. Thenis a symmetric {1}-inverse ofLG, where.According to Lemma 2.4, we could get the following results.
Lemma 2.6LetLGbe defined as above. If each row ofL1sums tok, then each column vector ofis proportional to the all-one vector, where.
ProofSuppose that the number of columns of L2 is n2 and let with ri being its i-th column vector, i = 1, 2, …, n2. First we show that for any ri, all the elements of are the same. If ri = 0, then the assertion holds since L1ri = 0. Otherwise, ri = −e. Since L1 is nonsingular with each row sum being k, it follows that each row of L−11 sums to . Thus , which also implies that all the elements of are the same. Hence, each column of is proportional to the all-one vector, that is, all the row vectors of are the same. It thus follows that each column of is proportional to the all-one vector, i.e. all the elements in any given column of are the same. □According to Lemma 2.6, we have the following result.
Lemma 2.7LetLGbe defined as above. If each row ofL1sums tok, then there exists a real numberξsuch that, where.
ProofLet . According to the argument in the proof of Lemma 2.6, all the row vectors in M1 are the same. On the other hand, since L1 is real symmetric, it follows thatLet . Then all the column vectors in M2 are the same since all the row vectors in are the same. Thus, we conclude that there exists a real number ξ such thatThis completes the proof. □
3 Main Results
In this section, we consider resistance distances between vertices in a specific subset S of V(G). Let S ⊂ V(G) such that all the vertices in S have the same neighborhood N in G − S. In the following, we give explicit formula for resistance distances between vertices in S. For simplicity, we use LS to denote the Laplacian matrix of the subgraph induced by S. Suppose that the cardinalities of S and N are n1 and k, respectively. Then the Laplacian matrix of G can be written as follows.where is the identity matrix of order n1.
Now we are ready to give formula for resistance distances between vertices in S.
LetS ⊂ V(G) such that all the vertices inShave the same neighborhoodNinG − S. Then fori, j ∈ S, we havewhere.
ProofLet . Clearly, L1 is nonsingular, and each row of L1 sums to k and each column vector of L2 is − e or 0. Then by Lemma 2.7, there exists a real number ξ such that , where . Then by Lemma 2.5, we can obtain the {1}-inverse of LG as follows.Thus, for vertices i, j ∈ S, by Lemma 2.3, we haveThe proof is complete. □
Theorem 3.1indicates that, if S ⊂ V(G) satisfies that all the vertices in S have the same neighborhood N in G − S, then resistance distances between vertices in S depends only on the subgraph G[S] and the cardinality of N. In other words, if we use G* to denote the subgraph obtained from G[S ∪ N] by deleting all the edges between vertices in N (see Figure 1), then resistance distances between vertices in S depends only on G*. In fact, for i, j ∈ S, , as shown in the following.
FIGURE 1
Illustration of graphs G and its subgraph G*.
LetS ⊂ V(G) such that all the vertices inShave the same neighborhoodNinG − S. LetG* the graph obtained fromG[S ∪ N] by deleting all the edges between vertices inN. Then fori, j ∈ S, we havewhere.
ProofAccording to the definition of G*, it is readily to see that the Laplacian matrix of G* isSince each column vector of is − e, by Lemma 2.5, we can obtain the symmetric {1}-inverse of as follows:where . Hence by Lemma 2.3, we haveas required. □
Remark 1Combining Theorems 3.1 and 3.2, we could conclude that if S ⊂ V(G) satisfies that all the vertices in S have the same neighborhood N in G − S, then resistance distances between vertices in S can be computed as in the subgraph obtained from G[S ∪ N] by deleting all the edges between vertices in N. It should be mentioned that this fact, known as the reduction principle, was established in [14]. We confirm this result by algebraic method, rather than electric network method as used in [14]. Furthermore, we also give an exact formula for resistance distances between vertices in S. By Theorem 3.1, we are able to establish some interesting properties.
LetS ⊂ V(G) such that all the vertices inShave the same neighborhoodNinG − S. Then fori, j ∈ Sandu ∈ G − S, we havewhere.
ProofAs given in the proof of Theorem 3.1, we know that the {1}-inverse of LG iswhere ξ be a real number and . By Lemma 2.3, we haveNote that L1 is nonsingular and every row sums to k and each column vector of L2 is − e or a zero vector. So by Lemma 2.6, we know that each column of is proportional to all-one vector, which implies that (X)iu = (X)ju. Since X is real symmetric, we also have (X)ui = (X)uj. It follows thatThis completes the proof. □It is interesting to note from Theorem 3.2 that the difference between ΩG (i, u) and ΩG (j, u) depends only on the subgraph G [S] and the cardinality of N, no matter the chosen of u. Then we have the following result.
Corollary 3.4LetS ⊂ V(G) such that all the vertices inShave the same neighborhoodNinG − S. Then fori, j ∈ Sandu, v ∈ G − S, we have
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
YY contributed to conception and design of the study. WS performed the theoretical analysis and wrote the first draft of the manuscript. YY revised the manuscript. Both authors read, and approved the submitted version.
Acknowledgments
The authors would like to thank the anonymous referees for their helpful comments and suggestions. The support of the National Natural Science Foundation of China (through Grant No. 12171414) and Natural Science Foundation of Shandong Province (through no. ZR2019YQ02) is greatly acknowledged.
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.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
resistance distance, Laplacian matrix, {1}-inverse, moore-penrose inverse, reduction principle
Citation
Sun W and Yang Y (2022) A Note on Resistance Distances of Graphs. Front. Phys. 10:896886. doi: 10.3389/fphy.2022.896886
Received
15 March 2022
Accepted
31 March 2022
Published
11 April 2022
Volume
10 - 2022
Edited by
Yongxiang Xia, Hangzhou Dianzi University, China
Reviewed by
Jia-Bao Liu, Anhui Jianzhu University, China
Audace A. V. Dossou-Olory, Université d'Abomey-Calavi, Benin
Updates
Copyright
© 2022 Sun and Yang.
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*Correspondence: Yujun Yang, yangyj@yahoo.com
This article was submitted to Social Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.