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This paper is concerned with the general atom-bond sum-connectivity index A B S γ , which is a generalization of the recently proposed atom-bond sum-connectivity index, where γ is any real number. For a connected graph G with more than two vertices, the number A B S γ (G) is defined as the sum of (1 − 2 (d x + d y) − 1) γ over all edges x y of the graph G, where d x and d y represent the degrees of the vertices x and y of G, respectively. For − 10 ≤ γ ≤ 10 , the significance of A B S γ is examined on the data set of twenty-five benzenoid hydrocarbons for predicting their enthalpy of formation. It is found that the predictive ability of the index A B S γ for the selected property of the considered hydrocarbons is comparable to other existing general indices of this type. The effect of the addition of an edge between two non-adjacent vertices of a graph under A B S γ is also investigated. Furthermore, several extremal results regarding trees, general graphs, and triangle-free graphs of a given number of vertices are proved. [ABSTRACT FROM AUTHOR]

Let G be a graph containing no isolated vertices. For the graph G, its modified first Zagreb index is defined as the sum of reciprocals of squares of vertex degrees of G. This article provides some new bounds on the modified first Zagreb index of G in terms of some other well-known graph invariants of G. From the obtained bounds, several known results follow directly. [ABSTRACT FROM AUTHOR]

Let G = (V;E), V = fv1; v2;:::; vng, be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d1 d2 dn > 0, di = d(vi). Let A = (aij)nn and D = diag(d1; d2;:::; dn) be the adjacency and the diagonal degree matrix of G, respectively. Denote by L+(G) = D-1=2(D + A)D-1=2 the normalized signless Laplacian matrix of graph G. The eigenvalues of matrix L+(G), 2 = + 1 + 2 = + n = 0, are normalized signless Laplacian eigenvalues of G. In this paper some bounds for the sum K+(G) = Pn i=1 1 + i are considered. [ABSTRACT FROM AUTHOR]

Let G be a non-trivial connected graph. If the vertices v i and v j of G are adjacent, we write i ∼ j , otherwise we write i ≁ j. Denote by d i the degree of the vertex v i of G. The modified second Zagreb index and coindex of G are defined as M 2 ∗ (G) = ∑ i ∼ j 1 d i d j and M 2 ∗ ¯ (G) = ∑ i ≁ j 1 d i d j , respectively. Several lower and upper bounds on the modified second Zagreb index/coindex are derived and all the graphs attaining these bounds are characterized. The obtained bounds can be used to deduce bounds for various other topological indices. [ABSTRACT FROM AUTHOR]

This study aimed to investigate the effect of 150 mg/L sodium fluoride (NaF) on redox status parameters and essential metals [copper (Cu), iron (Fe), and zinc (Zn)] in the blood, liver, kidney, brain, and spleen of Wistar rats and to determine the protective potential of selenium (Se) against fluoride (F-) toxicity. Male Wistar rats were randomly distributed in groups of five (n=5) receiving tap water (control) or water with NaF 150 mg/L, NaF 150 mg/L + Se 1.5 mg/L, and Se 1.5 mg/L solutions ad libitum for 28 days. Fluorides caused an imbalance in the redox and biometal (Cu, Fe, and Zn) status, leading to high superoxide anion (O2.-) and malondialdehyde (MDA) levels in the blood and brain and a drop in superoxide dismutase (SOD1) activity in the liver and its increase in the brain and kidneys. Se given with NaF improved MDA, SOD1, and O2.- in the blood, brain, and kidneys, while alone it decreased SH group levels in the liver and kidney. Biometals both reduced and increased F- toxicity. Further research is needed before Se should be considered as a promising strategy for mitigating F- toxicity. [ABSTRACT FROM AUTHOR]