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The harmonic polynomial was defined in order to understand better the harmonic topological index. Here, we obtain several properties of this polynomial, and we prove that several properties of a graph can be deduced from its harmonic polynomial. Also, we prove that graphs with the same harmonic polynomial share many properties although they are not necessarily isomorphic. [ABSTRACT FROM AUTHOR]

A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G/e obtained from the (simple or non-simple) graph G by contracting an arbitrary edge e from it. We prove that H is hyperbolic if and only if G is hyperbolic, for many minors H of G, even if H is obtained from G by contracting and/or deleting infinitely many edges. [ABSTRACT FROM AUTHOR]

If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics [ x 1 x 2 ] , [ x 2 x 3 ] and [ x 3 x 1 ] in X . The space X is δ - hyperbolic (in the Gromov sense) if any side of T is contained in a δ -neighborhood of the union of the two other sides, for every geodesic triangle T in X . We denote by δ ( X ) the sharpest hyperbolicity constant of X , i.e., δ ( X ) : = inf { δ ≥ 0 : X is δ -hyperbolic } . In the study of any parameter on graphs it is natural to study the graphs for which this parameter has small values. In this paper we study the graphs with small hyperbolicity constant, i.e., the graphs which are like trees (in the Gromov sense). We obtain simple characterizations of the graphs G with δ ( G ) = 1 and δ ( G ) = 5 4 (the case δ ( G ) < 1 is known). Also, we give a necessary condition in order to have δ ( G ) = 3 2 (we know that δ ( G ) is a multiple of 1 4 ). Although it is not possible to obtain bounds for the diameter of graphs with small hyperbolicity constant, we obtain such bounds for the effective diameter if δ ( G ) < 3 2 . This is the best possible result, since we prove that it is not possible to obtain similar bounds if δ ( G ) ≥ 3 2 . [ABSTRACT FROM AUTHOR]

Using the symmetry property of the inverse degree index, in this paper, we obtain several mathematical relations of the inverse degree polynomial, and we show that some properties of graphs, such as the cardinality of the set of vertices and edges, or the cyclomatic number, can be deduced from their inverse degree polynomials. [ABSTRACT FROM AUTHOR]