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The incidence energy I E (G) of a graph G, defined as the sum of the singular values of the incidence matrix of a graph G, is a much studied quantity with well known applications in chemical physics. The Laplacian-energy-like invariant of G is defined as the sum of square roots of the Laplacian eigenvalues. In this paper, we obtain the closed-form formulae expressing the incidence energy and the Laplacian-energy-like invariant of the Union Jack lattice. Moreover, the explicit asymptotic values of these quantities are calculated by utilizing the applications of analysis approach with the help of calculational software.

The computational complexity of the Hosoya index of a given graph is NP-Complete. Let [Formula: see text] be the graph constructed from [Formula: see text] by a triangle instead of all vertices of the initial graph [Formula: see text]. In this paper, we characterize the Hosoya index of the graph [Formula: see text]. To our surprise, it shows that the Hosoya index of [Formula: see text] is thoroughly given by the order and degrees of all the vertices of the initial graph [Formula: see text].

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Non-small cell lung cancer accounts for 85% of all types of lung cancer and is the leading cause of worldwide cancerassociated mortalities. MiR-124 is epigenetically silenced in various types of cancer and plays important roles in tumor development and progression. MiR-124 was also significantly downregulated in non-small cell lung cancer patients. Glycolysis has been considered as a feature of cancer cells; hypoxia-inducible factor 1-alpha/beta and Akt are key enzymes in the regulation of glycolysis and energy metabolism in cancer cells. However, the role of miR-124 in non-small cell lung cancer cell proliferation, glycolysis, and energy metabolism remains unknown. In this research, cell proliferation was investigated using 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide; furthermore, glucose consumption and lactic acid production were assessed; adenosine triphosphate content and NAD + /NADH were also detected. These tests were conducted using the normal non-small cell lung cancer cell line A549, which was transfected variedly with miR-mimics, miR-124 mimics, miR-124 inhibitor, pc-DNA3.1(+)-AKT1, and pc-DNA3.1(+)-AKT2 plasmid. Here, we show that miR-124 overexpression directly decreased cell growth, glucose consumption, lactate production, and energy metabolism. MiR-124 also negatively regulates glycolysis rate-limiting enzymes, glucose transporter 1 and hexokinase II. Our results also showed that miR-124 negatively regulates AKT1 and AKT2 but no regulatory effect on hypoxiainducible factor 1-alpha/beta. Overexpression of AKT reverses the inhibitory effect of miR-124 on cell proliferation and glycolytic metabolism in non-small cell lung cancer. AKT inhibition blocks miR-124 silencing-induced AKT1/2, glucose transporter 1, hexokinase II activation, cell proliferation, and glycolytic or energy metabolism changes. In summary, this study demonstrated that miR-124 is able to inhibit proliferation, glycolysis, and energy metabolism, potentially by targeting AKT1/2-glucose transporter 1/hexokinase II in non-small cell lung cancer cells.

The normalized Laplacian makes a great contribution on analyzing the structure properties of nonregular graphs. Let O n be a linear octagonal-quadrilateral network.In this article, we first concern the normalized Laplacian spectrum of O n based on the decomposition theorem for the corresponding matrices. Then we derive the closed-term formulas of the degree-Kirchhoff index and the number of spanning trees of linear octagonal-quadrilateral networks on the basis of the relations between the roots and coefficients, respectively. K E Y W O R D Sdegree-Kirchhoff index, normalized Laplacian, spanning tree

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