Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and μ(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that μ(D(G)) ≤ n(G)+1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with μ(D(G)) = n(G) + 1. An algorithm is given firstly to judge w...

In this paper, some new properties are presented to the extremal graphs with largest (signless Laplacian) spectral radii in the set of all the connected graphs with prescribed degree sequences, via which we determine all the extremal tricyclic graphs in the class of connected tricyclic graphs with prescribed degree sequences, and we also prove some majorization theorems of tricyclic graphs with...

The concept of geometric-arithmetic indices (GA) was put forward in chemical graph theory very recently. In spite of this, several works have already appeared dealing with these indices. In this paper we present lower and upper bounds on the second geometric-arithmetic index (GA2) and characterize the extremal graphs. Moreover, we establish Nordhaus-Gaddum-type results for GA2.

The Zagreb indices are the oldest graph invariants used in mathematical chemistry to predict the chemical phenomena. In this paper we define the multiple versions of Zagreb indices based on degrees of vertices in a given graph and then we compute the first and second extremal graphs for them.

Extremal graphs which are close related to Moore graphs have been defined in different ways. Radially Moore graphs are one of these examples of extremal graphs. Although it is proved that radially Moore graphs exist for radius two, the general problem remains open. Knor, and independently Exoo, gives some constructions of these extremal graphs for radius three and small degrees. As far as we kn...

Symmetric Division Deg index is one of the 148 discrete Adriatic indices that showed good predictive properties on the testing sets provided by International Academy of Mathematical Chemistry. Symmetric Division Deg index is defined by $$ SDD(G) = sumE left( frac{min{d_u,d_v}}{max{d_u,d_v}} + frac{max{d_u,d_v}}{min{d_u,d_v}} right), $$ where $d_i$ is the degree of vertex $i$ in graph $G$. In th...

eccentric connectivity index has been found to have a low degeneracy and hence a significantpotential of predicting biological activity of certain classes of chemical compounds. wepresent here explicit formulas for eccentric connectivity index of various families of graphs.we also show that the eccentric connectivity index grows at most polynomially with thenumber of vertices and determine the ...