Inspired by the chemical applications of higher-order connectivity index (or Randic index), we consider here the higher-order first Zagreb index of a molecular graph. In this paper, we study the linear regression analysis of the second order first Zagreb index with the entropy and acentric factor of an octane isomers. The linear model, based on the second order first Zag...

A retraction from a structure P to its substructure Q is a homomorphism from P onto Q that is the identity on Q. We present an algebraic condition which completely characterises all posets and all reflexive graphs Q with the following property: the class of all posets or reflexive graphs, respectively, that admit a retraction onto Q is first-order definable.

We first show how a special case of Jackson’s 8φ7 summation immediately gives Warnaar’s q-analogue of the sum of the first n cubes, as well as q-analogues of the sums of the first n integers and first n squares. Similarly, by appropriately specializing Bailey’s terminating very-well-poised balanced 10φ9 transformation and applying the terminating very-well-poised 6φ5 summation, we find q-analog...

We find Euler integral formulas, summation and reduction formulas for q-analogues of Srivastava’s three triple hypergeometric functions. The proofs use q-analogues of Picard’s integral formula for the first Appell function, a summation formula for the first Appell function based on the Bayley–Daum formula, and a general triple series reduction formula of Karlsson. Many of the formulas are purel...

Let S ⊂ Rk+m be a compact semi-algebraic set defined by P1 ≥ 0, . . . , P` ≥ 0, where Pi ∈ R[X1, . . . , Xk, Y1, . . . , Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ `. Let π denote the standard projection from Rk+m onto Rm. We prove that for any q > 0, the sum of the first q Betti numbers of π(S) is bounded by (k +m). We also present an algorithm for computing the the first q Betti numbers of π(S), whose com...