Spectra of quasi-corona and multiple Q-complemented-vertex join graphs

The Q-complemented graph of a graph $G$, denoted by $CT(G)$, was obtained by adding a new vertex for each edge $uv\in E(G)$, where this new vertex was adjacent to all vertices of $G$ except $u$ and $v$. In this paper, we introduced two new graph constructions based on $CT(G)$, namely, the quasi-corona Q-complemented-vertex join $G_1\odot G_2$ and the multiple Q-complemented-vertex join $G_1▽G_2$. We determined the adjacency, Laplacian, and signless Laplacian spectra of these composite graphs. Using these spectral results, we constructed infinite families of $A$-, $L$-, and $Q$-cospectral graphs. Moreover, we derived explicit formulas for several key invariants of the new graphs, including the number of spanning trees, the Kirchhoff index, the signless Laplacian energy-like invariant, and graph energy. Furthermore, numerical experiments were provided to illustrate the structural properties of the proposed graphs, suggesting that the multiple Q-complemented-vertex join construction may exhibit certain small-world-like characteristics.