On the Determination of Some Graphs by Resistance Spectra

Abstract

The resistance distance between two vertices in a connected graph is defined as the effective resistance between them when each edge of the graph is replaced by a unit resistor. The multiset of all pairwise resistance distances in a graph G constitutes its resistance spectrum, denoted RS(G). A graph G is said to be determined by its resistance spectrum (DRS) if, for any graph H satisfying

\mathit{RS}(H)=\mathit{RS}(G)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${RS}(H) = {RS}(G)$$\end{document}

, we have

H\cong G

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H \cong G$$\end{document}

. While computational studies reveal that the vast majority of small graphs are DRS, providing rigorous mathematical proofs for specific infinite families remains a challenging and active research area. This paper investigates the resistance spectral determination for four significant families of graphs: barbell graphs, chained silicate networks, Dutch windmill graphs, and caterpillar trees. By employing fundamental principles from electrical network theory—including the series and parallel rules, the cut-vertex principle, and Foster’s theorems—we explicitly analyze the resistance structure of these graphs. We then prove that each family is uniquely DRS. Our results contribute new infinite families of graphs to the growing catalog of those known to be DRS, thereby enhancing our understanding of the expressive power of the resistance spectrum in graph characterization.

Keywords

Resistance distance / Resistance spectrum / Graph characterization / Electrical network theory / Barbell graph / Dutch windmill graph / Caterpillar tree / Chained silicate network / 05C12 / 05C60

References

[1] Abdian AZ. Graphs which are determined by their spectrum. Konuralp J. Math., 2016, 4(2): 34-41

[2] Baxter, L.: Counterexample Wanted for Graph Isomorphism Conjecture. USENET: Comp. Theory (1999). Retrieved from https://groups.google.com/group/comp.theory/msg/e021f3199c528b63?dmode=source. Accessed 2 Sept 2024

[3] Cioabǎ SM, Haemers WH, Vermette JR, Wong W. The graphs with all but two eigenvalues equal to ±1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pm 1$$\end{document}. J. Algebraic Comb., 2015, 41(3): 887-897

[4] Collatz L, Sinogowitz U. Spektren endlieher Grafen. Abh. Math. Sem. Univ. Hamburg, 1957, 21: 63-77

[5] Diao Z, Pan XF, Liu XY. Analysis of some graphs through resistance spectra. Phys. Scr., 2025, 100 085254

[6] Foster, R.M.: The average impedance of an electrical network. In: Contributions to Applied Mechanics (Reissner Anniversary Volume). pp. 333–340. Edwards Brothers, Ann Arbor (1949)

[7] Ghosh, A., Boyd, S., Saberi, A.: Minimizing effective resistance of a graph. In: Proc. 17th Int. Symp. Math. Theory Net. Syst. (MTNS), Kyoto, Japan, July 24–28, pp. 1185–1196 (2006)

[8] Godsil, C.D., McKay, B.D.: Constructing cospectral graphs. Aequ. Math. 25, 257–268 (1982)

[9] Günthard, H.H., Primas, H: Zusammenhang von Graphentheorie und MO-Theorie von Molekeln mit Systemen konjugierter Bindungen. Helv. Chim. Acta 39(6), 1645–1653 (1956)

[10] Hückel, E.: Quantentheoretische Beiträge zum Benzolproblem. Z. Phys. 70, 204–286 (1931)

[11] Klein DJ. Resistance-distance sum rules. Croat. Chem. Acta, 2002, 75: 633-649

[12] Klein, D.J., Randić, M.: Resistance distance. J. Math. Chem. 12(1), 81–95 (1993)

[13] Li C, Bian H, Yu H, Cao Y. Resistance distance and Kirchhoff index based on degree for a cyclic triangular bipyramid hexahedron. Comput. Appl. Math., 2025, 44(1): 1-15

[14] Liu FJ, Huang Q. Laplacian spectral characterization of 3-rose graphs. Linear Algebra Appl., 2013, 439(10): 2914-2920

[15] Liu, J.B., Guan, L., Cao, J., Chen, L.: Coherence analysis for a class of polygon networks with the noise disturbance. IEEE Trans. Syst. Man Cybern: Syst. 55(7), 4718–4727 (2025)

[16] Liu, J.B., Wang, X., Cao, J.: The coherence and properties analysis of balanced 2p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2^p$$\end{document}-ary tree networks. IEEE Trans. Netw. Sci. Eng. 11(5), 4719–4728 (2024)

[17] Lu PL, Liu XG. Double starlike tree H(p,n,q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H(p, n, q)$$\end{document} determined by Laplacian spectrum. J. Harbin Eng. Univ., 2016, 37(02): 242-247

[18] Rickard, J.: Counterexample Wanted for Graph Isomorphism Conjecture. USENET: Comp. Theory (1999). Retrieved from https://groups.google.com/group/comp.theory/msg/53976638ed387d76?dmode=source. Accessed 2 Sept 2024

[19] Rosen A. A new network theorem. J. Inst. Electr. Eng., 1924, 62(335): 916-918

[20] Sardar MS, Liu JB, Siddique I, Jaradat MM. A novel and efficient method for computing the resistance distance. IEEE Access, 2021, 9: 107104-107110

[21] Sardar MS, Pan XF, Li YX. Some two-vertex resistances of the three-towers Hanoi graph formed by a fractal graph. J. Stat. Phys., 2020, 181(1): 116-131

[22] Sardar, M.S., Pan, X.F., Xu, S.J.: Computation of the resistance distance and the Kirchhoff index for the two types of claw-free cubic graphs. Appl. Math. Comput. 473, 128670 (2024)

[23] Sardar MS, Xu SJ. Resistance distance and Kirchhoff index in windmill graphs. Curr. Org. Synth., 2025, 22(2): 159-168

[24] Sardar, M.S., Xu, S.J., Pan, X.F.: Extremal values on the Kirchhoff index of the line graph of trees. Kuwait J. Sci. 52(1), 100327 (2025)

[25] Sardar, M.S., Zafar, S., Zahid, Z., Farahani, M.R., Wang, S., Naduvath, S.: Certain topological indices of line graph of Dutch windmill graphs. Southeast Asian Bull. Math. 44(1), 119–129 (2020)

[26] Strutt JW. On the theory of resonance. Philos. Trans. R. Soc. A, 1871, 161: 77-118

[27] Topcu H, Sorgun S. The kite graph is determined by its adjacency spectrum. Appl. Math. Comput., 2018, 330: 134-142

[28] Wang, J.F., Huang, Q.X., Belardo, F.: On the spectral characterizations of 3-rose graphs. Util. Math. 91, 33–46 (2013)

[29] Weisstein, E.W.: Resistance equivalent graphs (MathWorld—A Wolfram Web Resource) (2021). https://mathworld.wolfram.com/Resistance-EquivalentGraphs.html. Accessed 23 June 2025

[30] Wilson RC, Zhu P. A study of graph spectra for comparing graphs and trees. Pattern Recogn., 2008, 41(9): 2833-2841

[31] Xing, B., Sun, M., Zhou, H., Pan, X.F.: Determination of some graphs by resistance spectra. Comput. Appl. Math. 44(5), 237 (2025)

[32] Xu, S.-A.: Research on several kinds of graph parameters based on resistance distance. Dissertation. Anhui University, Hefei (2021). https://doi.org/10.26917/d.cnki.ganhu.2021.000561

[33] Zhang X, Zhang H. Some graphs determined by their spectra. Linear Algebra Appl., 2009, 431(9): 1443-1454

[34] Zhou, H., Ni, Q., Lyu, N.N., Pan, X.F.: Determination of some trees and unicyclic graphs by resistance spectra. J. Jimei Univ. Nat. Sci. 30, 286–291 (2025)

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