The augmented eccentric connectivity index of a graph which is a generalization of eccentric connectivity index is defined as the summation of the quotients of the product of adjacent vertex degrees and eccentricity of the concerned vertex of a graph. In this paper we established some relationships between augmented eccentric connectivity index and several other graph invariants like number of ...

Given a square, nonsingular matrix of univariate polynomials F ∈ K[x]n×n over a field K, we give a deterministic algorithm for finding the determinant of F. The complexity of the algorithm is O ̃(nωs) field operations where s is the average column degree or the average row degree of F. Here O ̃ notation is Big-O with log factors omitted and ω is the exponent of matrix multiplication.

We provide a short and self-contained proof of the classical result Kostochka Thomason, ensuring that every graph average degree d $d$ has complete minor order Ω ( ∕ log ) ${\rm{\Omega }}(d\unicode{x02215}\sqrt{{\rm{log}}d})$ .

An independent transversal of a graph G $G$ with vertex partition P ${\mathscr{P}}$ is an set intersecting each block in single vertex. Wanless and Wood proved that if has size at least t $t$ the average degree vertices most ∕ 4 $t\unicode{x02215}4$ , then exists. We present construction showing this result optimal: for any ε > 0 $\varepsilon \gt 0$ sufficiently large there forest into parts su...