A Spectral Study of Certain Graph Structures

Abstract

This Thesis aims to define new graph structures through graph operations and obtain the adjacency (A), Laplacian (L), and signless Laplacian (Q) spectra of these graphs. The spectra of newly constructed graph structures are analyzed, and these analyses are employed to determine various graph invariants, including Kirchhoff s index (Kf), spanning trees (t), and Laplacian energy-like invariant (LEL). We also obtain numerous collections of non-regular cospectral graphs and families of integral graphs, along with the finding of quasi-Laplacian energy (EQ), which is another aspect of our research. First, we define quasi-corona S-vertex join and quasi-corona S-edge join graphs. Following this, the spectra of A, L and Q matrices for these graphs are computed. Consequently, we identify numerous families of integral graphs, infinite collections of cospectral graphs, enumeration of the spanning trees (t), and the Kirchhoff s indices (Kf). Subsequently, two graphs quasi-corona R-vertex join and quasi-corona R-edge join are defined, followed by a calculation of the spectra for A, L and Q matrices associated with these graphs. As an application, we systematically identify infinite sets of graphs that share identical spectra, analyze Laplacian energy-like invariant (LEL) in detail, estimate the Kirchhoff s indices (Kf), and obtain the quantity of the spanning trees (t). Next, we define four novel graphs using join operation of central graph C and subdivision graph S. The graphs include the join of S vertex-C vertex, S edge-C edge, S edge-C vertex and S vertex-C edge. Following this, A and L-spectra of the four graphs are determined. The calculation of the Kirchhoff indices and the quantity of the spanning trees has been conducted. We analyze these four graphs A and L spectra and determine cospectral non-regular graphs by selecting two pairings of regular cospectral graphs. Finally, we delineate the connection between the quasi-Laplacian energy (EQ) of specific graphs and their corresponding original graphs.