CMMSE-on the first general Zagreb index

The aim of this paper is to obtain new inequalities involving the first general Zagreb index, and characterize graphs which are extremal with respect to them. We also obtain inequalities involving the forgotten and second general Zagreb indices.     

New upper bounds on Zagreb indices

The first Zagreb index M ₁(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M ₂(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper we obtain an upper bound on the first Zagreb index M ₁(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ₁), second maximum vertex degree (Δ₂) and minimum vertex degree (δ). Using this result we find an upper bound on M ₂(G). Moreover, we present upper bounds on and in terms of n,  m, Δ₁, Δ₂, δ, where denotes the complement of G.     

Some properties of the reformulated Zagreb indices

Miličević, Nikolić and Trinajstić (Mol Divers 8:393–399, 2004) proposed the reformed Zagreb indices in 2004. Now we give some properties for the reformed Zagreb indices.     

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.     

Exchangeable fraction of elements in alluvial sediments under waste disposal site (Zagreb, Croatia)

Concentrations of Ag, Ba, Cd, Ce, Cs, Co, Cr, Eu, Fe, Rb, Sc, Sr, Th, and Zn exchangeable fractions were determined in alluvial sediments at waste disposal site area in the vicinity of water-well field. Samples have been leached with 0.5M NH₄Cl at a sample/solution ratio of 120 during 24 hours without shaking. INAA of dry NH₄Cl residues show that the concentrations of exchangeable elements determined in the most of the sediments below the wastes have natural levels. Ag, Ba and Sr are readily exchangeable; Rb, Cs and Zn have lower exchangeability, while Cd, Ce, Th, Sc, Eu, Cr, Fe and Co are rather immobile. Extremely high total and exchangeable silver concentration was found at 6.5–6.8 meters below waste in the aerated layer occasionally under the water table. Exchangeable concentrations in deeper water-bearing sediment layers are not elevated. Due to this, one can presume that the upper sediment layers act as chemical filter generally preventing the infiltration from overlying wastes into water-bearing layers.