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The degree distance DD(G) of a connected graph G was invented by Dobrynin and Kochetova in 1994. Recently, one of the present authors introduced the concept of k-center Steiner degree distance defined as SDDk(G)=∑S⊆V(G)|S|=k[∑v∈Sdeg⁡G(v)]dG(S),SDD_k (G) = \sum\limits_{\mathop {S \subseteq V(G)}\limits_{\left| S \right| = k} } {\left[ {\sum\limits_{v \in S} {{\it deg} _G (v)} } \right]d_G (S),} where dG(S) is the Steiner k-distance of S and degG(v) is the degree of the vertex v in G. In this paper, we investigate the Steiner degree distance of complete and Cartesian product graphs.

The aim of this note is to establish new spectral bounds for the harmonic matrix.

Two decades ago, resistance distance was introduced to characterize “chemical distance” in (molecular) graphs. In this paper, we consider three resistance distance-based graph invariants, namely, the Kirchhoff index, the additive degree-Kirchhoff index, and the multiplicative degree-Kirchhoff index. Some Nordhaus-Gaddum-type results for these three molecular structure descriptors are obtained. In addition, a relation between these Kirchhoffian indices is established.

For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as , where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariants and named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = . The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

The atom-bond connectivity index (ABC) is a degree-based molecular structure descriptor with well-documented chemical applications. In 2010 a distance-based new variant of this index (ABCGG) has been proposed. Until now, the relation between ABC and ABCGG has not been analyzed. In this paper, we establish the basic characteristics of this relation. In particular, ABC and ABCGG are not correlated and both cases ABC > ABCGG and ABC < ABCGG may occur in the case of (structurally similar) molecules. However, in the case of benzenoid hydrocarbons, ABC always exceeds ABCGG. [Projekat Ministarstva nauke Republike Srbije, br. 174033]

The paper gives a short survey of the most important lower and upper bounds for total π-electron energy, i.e., graph energy (E). In addition, a new lower and a new upper bound for E are deduced, valid for general molecular graphs. The strengthened versions of these estimates, valid for alternant conjugated hydrocarbons, are also reported.