Uncertain graphs are employed to describe graph models with in-deterministic information that produced by human beings. This paper aims to study the maximum matching problem in uncertain graphs. The number of edges of a maximum matching in a graph is called matching number of the graph. Due to the existence of uncertain edges, the matching number of an uncertain graph is essentially an uncertain variable. Different from that in a deterministic graph, it is more meaningful to investigate the uncertain measure that an uncertain graph is k-edge matching (i.e., the matching number is greater than or equal to k). We first study the properties of the matching number of an uncertain graph, and then give a fundamental formula for calculating the uncertain measure. We further prove that the fundamental formula can be transformed into a simplified form. What is more, a polynomial time algorithm to numerically calculate the uncertain measure is derived from the simplified form. Finally, some numerical examples are illustrated to show the application and efficiency of the algorithm.

In this paper, it is shown that when a special vertex stretching is applied to a graph, the cochordal cover number of the graph in-creases at most two. As a consequence, it is shown that the induced matching number and cochordal cover number of a special vertex stretching of a graph G are equal provided G is well-covered bipartite or weakly chordal graph.

The traditional method for studying non-stationary signals is spectrogram based on the short-time Fourier transform (STFT). The well known limitation of the STFT is the inherent trade-off between time and frequency resolution. The Wigner-Ville (WV) distribution has the best time-frequency resolution, but its draw back is generating cross-terms. The matching pursuit (MP) distribution based on using the Gaussian atom is always positive, does not include crossterm, and has convenient resolution. In this paper, we have shown in addition to the known properties, the MP distribution can also remove the additive noise inherently. On the other words, we are able to remove the noise just by limiting the algorithm iterations and without paying any additional cost. Although the MP distribution based on using the Gaussian atoms is always positive and it has convenient resolution, according to the MP the time marginal and the frequency marginal will not be obtained accurately. In this paper, it has been shown that by implementing the minimum cross entropy (MCE) technique according to the MP distribution as a priory positive distribution, the new extracted distribution has the most similarity to the MP distribution and it also satisfies the correct time and frequency marginal.

The modified first Zagreb connection index ZC∗1 for a graph G is defined as ZC∗1 (G) = \sum v∈V (G) dvτv , where dv is the degree of the vertex v and τv denotes the connection number of v (that is, the number of vertices at the distance 2 from the vertex v). Let Tn,α be the class of trees with order n and matching number α such that n > 2α−1. In this paper, we obtain the lower bounds on the modified first Zagreb connection index of trees belonging to the class Tn,α, for 2α − 1 < n < 3α + 2.

A perfect star packing in a fullerene graph G is a spanning subgraph of G whose every component is isomorphic to the star graph K_1,3. A perfect pseudo matching of a fullerene graph G is a spanning subgraph H of G such that each component of H is either K_2 or K_1,3. In this paper, we examine the number of perfect star packing in (3,6)-fullerene graphs and perfect pseudo matching in chamfered fullerene graphs.

‎The multiplicative first Zagreb index is the product of the square of the degree of vertices in a graph $\mathbb{G}$‎. ‎The multiplicative reformulated first Zagreb index is defined as $\prod_{1,e}(\mathbb{G})= \prod_{x_{1}x_{2}\in E(\mathbb{G})}(d_{\mathbb{G}}(x_{1})+d_{\mathbb{G}}(x_{1})-2)^{2}$‎, ‎where $E(\mathbb{G})$ is the edge set of a graph $\mathbb{G}$ and $d_{\mathbb{G}}(x_{1})$ is the degree of a vertex $x_{1}$ in a graph $\mathbb{G}$‎. ‎In this paper‎, ‎we characterize the minimum and maximum trees and unicyclic graphs with respect to matching and perfect matching using this graph invariant $\prod_{1,e}(\mathbb{G})$ among the collection of all $n$-vertex graphs‎.

The extremal Gutman index is a concept that studies the maximum or minimum value of the Gutman index for a particular class of graphs. This research area is concerned with finding the graphs that have the lowest possible Gutman index within a set of graphs that have been transformed in some way, such as by adding or removing edges or vertices. By understanding the graphs that have the lowest possible Gutman index, researchers can better understand the fundamental principles of graph stability and the role that different graph transformations play in affecting the overall stability of a graph. The research in this area is ongoing and continues to expand as new techniques and algorithms are developed. The findings from this research have the potential to have a significant impact on a wide range of fields and can lead to new and more effective ways of analyzing and understanding complex systems and relationships in a variety of applications. This paper focuses on the study of specific types of trees that are defined by fixed parameters and characterized based on their Gutman index. Specifically, we explore the structural properties of graphs that have the lowest Gutman index within these classes of trees. To achieve this, we utilize various graph transformations that either decrease or increase the Gutman index. By applying these transformations, we construct trees that satisfy the desired criteria.

< p >The various subject matters of jurisprudence (Fiqh) contain many religious laws limited by legal measures. In these cases, the correct recognition of the fulfillment of legal measure and the way it is performed by religiously competent person is an important matter. Four theories can be driven from the expressions of the jurists in this regard including rational precision, customary precision, customary negligence and statement in detail. When it is the matter of limitation, the customary precision is the criterion for recognition of duty, but considering the way of its performance the customary accuracy is acceptable. In addition, considering the recognition of the concept of measure, the customary negligence is considered to be harmless, but considering adapting an example to the concepts, it is not acceptable. Alongside these matters, scientific and experimental precision can also be studied. While quoting and criticizing the statements, this article prefers the statement in detail. In most cases, there is a deliberation and insistence on strict observance of the specified limits in the word of divine law which indicates the importance of a particular amount in the view of the Legislator and the tolerance thereof is rejected. However, in some cases, the method of confinement may be discovered from the language of reason or other evidences and that the substantiation of a particular amount is the detector of the actual realization of the subject of religious law. Therefore, in those cases, accuracy is not demanded.