EXECUTIVE SUMMARY OF UGC MINOR RESEARCH PROJECT


 

Principal Investigator: Ms. Nutan G. Nayak

Department: Mathematics and Statistics

Title of the Project: Study of Spectra and Energy of Signed Graphs

Duration: 2 years (6th April 2015 – 6th April 2017)

Total allocated amount for the project: Rs. 1, 80, 000 /-

 

Objectives of the Project:
  • To determine spectra and energy of various signed graphs like regular, bipartite, complete signed graphs, etc.
  • To characterize signed graphs which are equienergetic and hyperenergetic.

Methodology:

  • Studied the existing results related to the topic.
  • Modified the existing results using linear algebra.
  • Developed new results for signed graphs.
  • For the research carried out, MATLAB and other software were used for computation.

Summary of the Findings:

1. Report of the work done in the first year:

In this study, spectra for heterogeneous unbalanced signed graphs have been developed. Net-regular signed graphs are characterized and proved that there exists a net-regular signed graph on every regular graph but the converse does not hold good. A family of connected, net-regular signed graphs is constructed whose underlying graphs are not regular and the spectra for these signed graphs are established. Further, the spectrum for one class of heterogeneous unbalanced net-regular signed complete graphs is obtained.

Paper Published:

  • Paper titled ‘On Net Regular Signed Graphs’ published in International Journal of Mathematical Combinatorics, 1(2016), 57-64.

Paper Presented:

  • Presented a paper titled ‘On Net Regular Signed Graphs’ at the International Conference on Mathematical Sciences (ICMS-2015) organized by Sri Venkateswara University, Tirupati, Andra Pradesh during 13th to 15th July 2015.
  • Presented a paper titled ‘Equienergetic Signed Bipartite Graphs’ at the International Conference on Mathematics, Physics and Allied Sciences organized by Carmel College Nuvem, Goa and in collaboration with International Multidisciplinary Research Foundation during 3rd to 5th March, 2016.

2. Report of the work done in the second year:

In this work, it is shown that there exists an infinite family of signed graphs having maximum signed energy in the class of signed graphs which are satisfying pairing property. Also, proved that it is possible to compare the energies of a pair of bipartite and non-bipartite signed graphs which belong to the class of pairing property on n vertices by defining quasi-order relation in such a way that the energy is increasing.

The concept of extended double cover of graphs is extended to signed graphs in order to establish the spectra of various unbalanced signed bipartite graphs. Then, non-cospectral equienergetic signed bipartite graphs on 4n vertices are constructed.

Further, net-Laplacian matrix is defined by considering the edge signs of a signed graph and the relation of net-Laplacian eigenvalues are obtained. Bounds for signed net-Laplacian eigenvalues as well as signed graph eigenvalues are established.

Moreover, net-Laplacian energy of a signed graph is introduced and proved that net-Laplacian energy equals signed energy if a given signed graph is net-regular. All signed graphs which are equienergetic for their Adjacency, Laplacian and net-Laplacian matrices are characterized. Upper and lower bounds for net-Laplacian energy are also established. An example of a pair of unbalanced, non-cospectral, net-Laplacian equienergetic signed graph is given.

Paper Published:

  • Paper titled ‘On net-Laplacian Energy of Signed Graphs’ accepted for publication in Communications in Combinatorics and Optimization(in press).
  • Paper titled ‘Spectra and Energy of Signed Graphs’ published in International Journal of Mathematical Combinatorics , 1(2017), 10-21.

Paper Presented:

  • Presented a paper titled ‘On some Inequalities of net-Laplacian Energy of Signed Graphs’ at the Silver Jubilee International Conference on Interdisciplinary Mathematics, Statistics and Computational Techniques(IMSCT 2016-FIMXXV) organized by Manipal University, Jaipur during 22nd to 24th December, 2016.