Embedding of sign-regular signed graphs and its spectral analysis

ABSTRACT

A $signedgraph$ is a graph whose edges carry the weight ‘+’ or ‘−’. A signed graph S is called sign-regular if $d^{-}\left(v\right)$ is same for all $v\in V$ and $d^{+}\left(v\right)$ is same for all $v\in V$. The problems of embedding $(i,j)$-sign-regular signed graphs in $(i+k,j+l)$-sign-regular signed graphs is one of the fascinating problem from application point of view which is dealt in this paper with insertion of least number of vertices in S and the problem of finding least number of non-isomorphic co-regular signed graphs is explored. We also define the relationship between characteristic polynomial of graph G and the graph in which it embeds.

Keywords:

• Signed graph
• sign-regular signed graph
• embedding
• co-regular signed graphs
• characteristic polynomial

2010 Mathematics Subject Classifications:

• 05C22
• 05C60

Acknowledgments

The authors are very much thankful to Dr B. D. Acharya for motivating them to work on more results on this idea of embedding of signed graphs with different aspect other than regularity but could not witness its growth. This work is supported by the Research Grant from DST [MTR/2018/000607] under Mathematical Research Impact Centric Support (MATRICS) for a period of 3-years (2019–2022) and University Grants Commission [Sr. No. 2061540883 Ref. No. $21/06/2015\left(i\right)EU-V$] of first and second authors, respectively.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the Research Grant from DST, [MTR/2018/000607] under Mathematical Research Impact Centric Support (MATRICS) for a period of 3-years (2019–2022) and University Grants Commission [Sr. No. 2061540883 Ref. No. $21/06/2015\left(i\right)EU-V$] of first and second authors, respectively.