Search

The general sum-connectivity index of a graph G is defined as $$\chi _\alpha (G)=\sum _{uv\in E(G)}(d(u)+d(v))^{\alpha }$$, where d(u) denotes the degree of a vertex u in G and $$\alpha $$ is a real number. In this paper, we determine the minimum general sum-connectivity indices of trees with n vertices and matching number m, where $$n=2m$$ for $$\alpha \le -\,2$$ and $$2m\le n\le 3m+1$$ for $$\alpha >1$$, respectively. The corresponding extremal graphs are also characterized.

The acrylate photosensitive resin hybrids containing 2-hydroxyethyl methacrylate-functionalized graphene oxide (HEMA-g-GO/PRFRPCP) were prepared. 2-Hydroxyethyl methacrylate-grafted graphene oxide (HEMA-g-GO) was synthesized by in situ covalent functionalization of graphene oxide (GO) with 2-hydroxyethyl methacrylate (HEMA). The effects of HEMA-g-GO on the properties of the photosensitive resin were evaluated through evaluating the variation in viscosity and gel rate. Compared with GO, HEMA-g-GO addition doesn’t more markedly increase the viscosity and also doesn’t reduce the gel rate until the addition amount of HEMA-g-GO reaches 0.06 wt% in the resin. The best result of 3.6% volume shrinkage for the nanocomposite cured with UV irradiation from the resin hybrid HEMA-g-GO/PRFRPCP containing 0.06 wt% HEMA-g-GO was obtained. The nanocomposites show the enhancement in mechanical properties and thermal properties even with very small HEMA-g-GO additions (0.01–0.06 wt%). Furthermore, the behaviors of the nanocomposites under dynamic thermal conditions were examined with dynamic thermomechanical analysis, confirming that there is the excellent interfacial adhesion between HEMA-g-GO and the resin matrix which is responsible for the enhanced properties of the nanocomposites.

The general Randic index of a graph G is defined as R α ( G ) = ź u v ź E ( G ) ( d ( u ) d ( v ) ) α , where d(u) denotes the degree of a vertex u in G and α is a real number. In this paper, we determine the maximum general Randic indices of trees and chemical trees with n vertices and k pendent vertices for 4 ź k ź ź n + 2 3 ź and α0 ź α < 0, where α 0 ź - 0.5122 is the unique non-zero root of the equation 6 ź 4 α - 20 ź 9 α + 10 ź 12 α - 16 α + 5 ź 24 α = 0 . The corresponding extremal graphs are also characterized.

The general sum-connectivity index of a graph G is defined as (G)=uvE(G)(d(u)+d(v)), where d(u) denotes the degree of a vertex u in G and is a real number. In this paper, we present the maximum general sum-connectivity indices of trees and chemical trees with given number of pendent vertices for 1

The atom-bond connectivity ( A B C ) index of a graph G is defined as the sum of the weights d ( u ) + d ( v ) − 2 d ( u ) d ( v ) of all edges u v of G , where d ( u ) denotes the degree of a vertex u in G . In this short note, we determine the maximum A B C index for graphs with edge-connectivity one and characterize the corresponding extremal graphs. This answers a recent question proposed by Zhang et al. (2016).

The harmonic index of a graph G is defined as the sum of the weights 2/d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this work, we present the minimum, second-minimum, maximum and second-maximum harmonic indices for unicyclic graphs with given girth, and characterize the corresponding extremal graphs.

The atom-bond connectivity (ABC) index is a much-studied molecular structure descriptor, based on the degrees of the vertices of the molecular graph. Recently, another vertex-degree-based topological index - the harmonic index (H) - attracted attention and gained popularity. We show how ABC and H are related.

Abietic acid was isolated from rosin by means of reaction-crystallization coupled with exposure to ultrasonic waves, and it was then characterized by its specific rotation, infrared spectra, and 13C nuclear magnetic resonance spectra. The thermal decomposition of abietic acid in argon atmosphere was studied under nonisothermal conditions using TG-DTG techniques with heating rates of 5, 10, 20, and 25 K/min. For the kinetic study, the nonisothermal kinetic parameters were obtained via the analysis of the TG-DTG curves by using the Flynn-Wall-Ozawa (FWO) method, the Kissinger method, and the integral method. The results showed that the nonisothermal decomposition mechanism of abietic acid in argon atmosphere followed Mampel Power law with n = 3/2, whose differential and integral forms were f(α) = 2/3α–1/2 and G(α) = α3/2. The apparent activation energy Ea and the pre-exponential factor A were 123.44 kJ/mol and 1.78 × 1011 s–1, respectively. The kinetic equation can be expressed as dα/dt = 1.19 × 1011α–1/2 e...

The general Randic index R"@a(G) of a graph G is defined by R"@a(G)[email protected]?"u"v(d(u)d(v))^@a, where d(u) is the degree of a vertex u, and the summation extends over all edges uv of G. Some results on trees with a given order and matching number that have minimum general Randic index have been obtained. However, the corresponding maximum problem has not been studied, and usually the maximum problem is much harder than the minimum one. In this paper, we characterize the structure of the trees with a given order and matching number that have maximum general Randic index for @a>1 and give a sharp upper bound for 0

The Hosoya index and the Merrifield–Simmons index of a graph are defined as the total number of the matchings (including the empty edge set) and the total number of the independent vertex sets (including the empty vertex set) of the graph, respectively. Let V n , k be the set of connected n -vertex graphs with connectivity at most k . In this note, we characterize the extremal (maximal and minimal) graphs from V n , k with respect to the Hosoya index and the Merrifield–Simmons index, respectively.