Abstract
Let G = (V,E), V = {1, 2,…,n}, be a simple connected graph with n vertices and m edges and let d1 ≥ d2 ≥⋅ ⋅⋅≥ dn > 0, be the sequence of its vertex degrees. With i ∼ j we denote the adjacency of the vertices i and j in G. The inverse sum indeg index is defined as ISI = ∑ -didj- di+dj with summation going over all pairs of adjacent vertices. We consider lower bounds for ISI. We first analyze some lower bounds reported in the literature. Then we determine some new lower bounds.
Highlights
Let G = (V, E), V = {1, 2, . . . , n}, E = {e1, e2, . . . , em}, be a simple connected graph with n vertices and m edges, and let ∆ = d1 ≥ d2 ≥ · · · ≥ dn = δ > 0, di = d(i), and d(e1) ≥ d(e2) ≥ · · · ≥ d(em), be sequences of its vertex and edge degrees, respectively
We denote by ∆e1 = d(e1) + 2 and δe1 = d(em) + 2
An invariant is a property of graphs that depends only on their abstract structure, not on the labeling of vertices or edges, or on the drawing of the graph
Summary
It should be noted that Π∗1, SCI, and H can be considered as edge–degree based topological indices as well, since the following identities hold: m In a series of papers [26,27,28, 31], Vukičević introduced the so-called Adriatic indices, providing a general method for constructing vertex–degree based graph invariants; for review see [29]. In [5] it was proven m2 ISI ≥ , n with equality if and only if the graph G is regular or biregular.
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