The Zagreb indices have been introduced by Gutman and Trinajstić as M1(G) = v∈V (G) (dG(v)) 2 and M2(G) = uv∈E(G) dG(u)dG(v), where dG(u) denotes the degree of vertex u. We now define a new version of Zagreb indices as M * 1 (G) = uv∈E(G) [εG(u) + εG(v)] and M * 2 (G) = uv∈E(G) εG(u)εG(v), where εG(u) is the largest distance between u and any other vertex v of G. The goal of this paper is to further the study of these new topological index.

The quantum integrability of the 1D ionic Hubbard model (IHM) is established using two independent numerical methods, namely i) energy level spacing statistics and ii) occupation profile of one‐particle density matrix (OPDM) eigen‐values. Both methods suggest that the 1D IHM is integrable. The calculations of energy level statistics reproduce the known results for the standard Hubbard model. Upon turning on the the ionic term, the energy level spacing distribution of this model continues to obey the Poissonian distribution. Occupation patterns as extracted from OPDM indicate that quasi‐particles are sharpened upon increasing the ionic potential. This is evidenced by a larger jump in the occupation number distribution.

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