Open AccessUpper and lower bounds for the mixed degree-Kirchhoff index
Author(s) -
Monica Bianchi,
Alessandra Cornaro,
José Palacios,
Anna Torriero
Publication year - 2016
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil1609351b
Subject(s) - mathematics , combinatorics , degree (music) , inverse , upper and lower bounds , vertex (graph theory) , graph , order (exchange) , discrete mathematics , mathematical analysis , geometry , physics , finance , acoustics , economics
We introduce the mixed degree-Kirchhoff index, a new molecular descriptor defined by R^(G) = ?i<j(di/dj+dj/di)Rij, where di is the degree of the vertex i and Rij is the effective resistance between vertices i and j. We give general upper and lower bounds for bR(G) and show that, unlike other related descriptors, it attains its largest asymptotic value (order n4), among barbell graphs, for the highly asymmetric lollipop graph. We also give more refined lower (order n2) and upper (order n3) bounds for c-cyclic graphs in the cases 0 ? c ? 6. For this latter purpose we use a close relationship between our new mixed degree-Kirchhoff index and the inverse degree, prior bounds we found for the inverse degree of c-cyclic graphs, and suitable expressions for the largest and smallest effective resistances of c-cyclic graphs.