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The degree distance DD(G) of a connected graph Gwas 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 Sand degG(v) is the degree of the vertex vin 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 vertex PI index is a distance-based molecular structure descriptor, that recently found numerous chemical applications. Lower and upper bounds for PI are obtained, as well as results of Nordhaus-Gaddum type. Also a relation between the Szeged and vertex PI indices is established

Let Gbe a simple graph of order nwith Laplacian spectrum {λn, λn−1, …, λ1} where 0=λn≤λn−1≤⋅≤λ1. If there exists a graph whose Laplacian spectrum is S={0, 1, …, n−1}, then we say that Sis Laplacian realizable. In 6, Fallat et al. posed a conjecture that Sis not Laplacian realizable for any n≥2 and showed that the conjecture holds for n≤11, nis prime, or n=2, 3(mod4). In this article, we have proved that (i) if Gis connected and λ1=n−1 then Ghas diameter either 2 or 3, and (ii) if λ1=n−1 and λn−1=1 then both Gand Ḡ, the complement of G, have diameter 3. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 106–113, 2010