We review basic mathematical properties of the augmented eccentric connectivity index. Explicit formulas are presented for several classes of graphs, in particular for some open and closed unbranched polymers and nanostructures. Asymptotic behavior is explored and compression ratios are computed for those polymers.
The eccentric connectivity index ξ(G) of the graph G is defined as ξ(G) = Σu∈V(G) deg(u)ε(u) where deg(u) denotes the degree of vertex u and ε(u) is the largest distance between u and any other vertex v of G. In this paper an exact expression for the eccentric connectivity index of an armchair polyhex nanotube is given.
A signed graph consists of a (simple) graph G = (V, E) together with a function σ : E → {+, −} called signature. Matrices can be associated to signed graphs and the question whether a signed graph is determined by the set of its eigenvalues has gathered the attention of several researchers. In this paper we study the spectral determination with respect to the Laplacian spectrum of signed ∞-graphs. After computing some spectral invariants and obtain some constraints on the cospectral mates, we obtain some non isomorphic signed graphs cospectral to signed ∞-graphs and we study the spectral characterization of the signed ∞-graphs containing a triangle.
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