Spectral properties of the atom-bond sum-connectivity matrix

The atom-bond sum-connectivity index (ABS index) is a topological index introduced in 2022.For a graph G, this index is defined aswhere di is the degree of a vertex vi of G and E(G) is the set of edges of G.The ABS-matrix is defined as S(G) = [aij] n×n , where aij equals (di + dj -2)/(di + dj) when vivj ∈ E(G) and aij = 0 otherwise.Furthermore, the Laplacian ABS-matrix is defined ask=1 a ik when i = j and dij = 0 when i = j.In this paper, we first present several bounds on the ABS index.We then explore several properties of the eigenvalues of the ABS-matrix and Laplacian ABS-matrix.Finally, inspired by the definition of the convex linear combination of the adjacency matrix and diagonal matrix proposed by Nikiforov, we also define the convex linear combination of the ABS-matrix and ABS-diagonal matrix, and present some of its fundamental properties.