Nano-Zagreb Index and Multiplicative Nano-Zagreb Index of Some Graph Operations

Let G be a graph with vertex set V(G) and edge set E(G). The Nano-Zagreb and multiplicative Nano-Zagreb indices of G are NZ(G) = \prod_{uv \in E(G)} (d^2(u) - d^2(v)) and N*Z(G) = \prod_{uv \in E(G)} (d^2(u) - d^2(v)), respectively, where d(v) is the degree of the vertex v. In this paper, we define two types of Zagreb indices based on degrees of vertices. Also the Nano-Zagreb index and multiplicative Nano-Zagreb index of the Cartesian product, symmetric difference, composition and disjunction of graphs are computed

New Bounds for the Harmonic Energy and Harmonic Estrada index of Graphs

Let $G$ be a finite simple undirected graph with $n$ vertices and $m$ edges. The Harmonic energy of a graph $G$, denoted by $\mathcal{H}E(G)$, is defined as the sum of the absolute values of all Harmonic eigenvalues of $G$. The Harmonic Estrada index of a graph $G$, denoted by $\mathcal{H}EE(G)$, is defined as $\mathcal{H}EE=\mathcal{H}EE(G)=\sum_{i=1}^{n}e^{\gamma_i},$ where $\gamma_1\geqslant \gamma_2\geqslant \dots\geqslant \gamma_n$ are the $\mathcal{H}$-$eigenvalues$ of $G$. In this paper we present some new bounds for $\mathcal{H}E(G)$ and $\mathcal{H}EE(G)$ in terms of number of vertices, number of edges and the sum-connectivity index

Extremal trees for the Randić index

Graph theory has applications in various fields due to offering important tools such as topological indices. Among the topological indices, the Randić index is simple and of great importance. The Randić index of a graph d4a2; can be expressed as R ( G ) = ∑ x y ∈ Y ( G ) 1 τ ( x ) τ ( y ) Rłeft( G \right) = \sum\nolimitsxy \in Yłeft( G \right) 1 øver \sqrt τ łeft( x \right)τ łeft( y \right) , where d4b4;(d4a2;) represents the edge set and τ(x) is the degree of vertex x. In this paper, considering the importance of the Randić index and applications two-trees graphs, we determine the first two minimums among the two-trees graphs

Antioxidant activity and ACE-inhibitory of Class II hydrophobin from wild strain Trichoderma reesei

International audienceThere are several possible uses of the Class II hydrophobin HFBII in clinical applications. To fully understand and exploit this potential however, the antioxidant activity and ACE-inhibitory potential of this protein need to be better understood and have not been previously reported. In this study, the Class II hydrophobin HFBII was produced by the cultivation of wild type Trichoderma reesei. The crude hydrophobin extract obtained from the fermentation process was purified using reversed-phase liquid chromatography and the identity of the purified HFBII verified by MALDI-TOF (molecular weight: 7.2 kDa). Subsequently the antioxidant activities of different concentrations of HFBII (0.01–0.40 mg/mL) were determined. The results show that for HFBII concentrations of 0.04 mg/mL and upwards the protein significantly reduced the presence of ABTS+ radicals in the medium, the IC50 value found to be 0.13 mg/mL. Computational modeling highlighted the role of the amino acid residues located in the conserved and exposed hydrophobic patch on the surface of the HFBII molecule and the interactions with the aromatic rings of ABTS. The ACE-inhibitory effect of HFBII was found to occur from 0.5 mg/mL and upwards, making the combination of HFBII with strong ACE-inhibitors attractive for use in the healthcare industry

Lower bounds for the energy of graphs

Let G be a finite simple undirected graph with n vertices and m edges. The energy of a graph G , denoted by E ( G ) , is defined as the sum of the absolute values of the eigenvalues of G . In this paper we present lower bounds for E ( G ) in terms of number of vertices, edges, Randić index, minimum degree, diameter, walk and determinant of the adjacency matrix. Also we show our lower bound in (11) under certain conditions is better than the classical bounds given in Caporossi et al. (1999), Das (2013) and McClelland (1971). Keywords: Energy (of non-singular graph), Determinant of adjacency matrix, Randić index, Spectral radiu

The Behavior of Weighted Graph’s Orbit and Its Energy

Studying the orbit of an element in a discrete dynamical system is one of the most important areas in pure and applied mathematics. It is well known that each graph contains a finite (or infinite) number of elements. In this work, we introduce a new analytical phenomenon to the weighted graphs by studying the orbit of their elements. Studying the weighted graph's orbit allows us to have a better understanding to the behaviour of the systems (graphs) during determined time and environment. Moreover, the energy of the graph’s orbit is given

On the Spectrum of Laplacian Matrix

Let G be a simple graph of order n. The matrix ℒG=DG−AG is called the Laplacian matrix of G, where DG and AG denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let l1G, ln−1G be the largest eigenvalue, the second smallest eigenvalue of ℒG respectively, and λ1G be the largest eigenvalue of AG. In this paper, we will present sharp upper and lower bounds for l1G and ln−1G. Moreover, we investigate the relation between l1G and λ1G

New Results on Zagreb Energy of Graphs

Let G be a graph with vertex set VG=v1,…,vn, and let di be the degree of vi. The Zagreb matrix of G is the square matrix of order n whose i,j-entry is equal to di+dj if the vertices vi and vj are adjacent, and zero otherwise. The Zagreb energy ZEG of G is the sum of the absolute values of the eigenvalues of the Zagreb matrix. In this paper, we determine some classes of Zagreb hyperenergetic, Zagreb borderenergetic, and Zagreb equienergetic graphs

Spectral Properties with the Difference between Topological Indices in Graphs

Let G be a graph of order n with vertices labeled as v1,v2,…,vn. Let di be the degree of the vertex vi, for i=1,2,…,n. The difference adjacency matrix of G is the square matrix of order n whose i,j entry is equal to di+dj−2−1/didj if the vertices vi and vj of G are adjacent or vivj∈EG and zero otherwise. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix, that is, a modification of the classical adjacency matrix involving the degrees of the vertices. In this paper, some properties of its characteristic polynomial are studied. We also investigate the difference energy of a graph. In addition, we establish some upper and lower bounds for this new energy of graph