A unified criterion for distinguishing graphs by their spectral radius

Abstract

Complementarity spectrum of a connected graph G, denoted by $\mathrm{\Pi }(G)$, is the set of spectral radii of all connected induced subgraphs of G. Further, G is said to be spectrally non-redundant if $c(G)$, the cardinality of $\mathrm{\Pi }(G)$, is equal to $b(G)$, the number of all non-isomorphic induced subgraphs of G. In this paper, we give a sufficient condition for a family of graphs to be spectrally non-redundant. Using this criterion, we show that several infinite families of graphs are spectrally non-redundant. Moreover, we apply the same condition to distinguish graphs by their spectral radius, which illustrates the main reason for associating a graph with its complementarity spectrum.

Keywords:

• Spectral radius
• complementarity eigenvalues
• induced subgraph
• largest eigenvalue

2020 Mathematics Subject Classifications:

• 05C50
• 05C05
• 15A18

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of Pawan Kumar is supported by CSIR, India, as a JUNIOR RESEARCH FELLOWSHIP, file No. 09/112(0669)/2020-EMR-I. The research of Prof. S. Pirzada is supported by the SERB-DST research project number CRG/2020/000109. Vilmar Trevisan acknowledges partial support MATH-AMSUD under project GSA, Brazilian team financed by CAPES under project 88881.694479/2022-01, CNPq [grant numbers 409746/2016-9 and 303334/2016-9], and FAPERGS [grant number PqG 17/2551-0001].