The right path is a target and destination for which human being prays God for the relevant training and direction. It is a path which being in line with constitutes the destination and real entity and PERFECTion of mankind. The present paper is the result of a research on the verses of Koran that proves the real mystical MATCHING of a PERFECT man (same as chaste Imam in Shiite thinking) with the right path without direct reference to mystical ideas. To this end, the interpretation of the Koran after the Koran, as well as contemplation over the verses documented by the library sources are applied. In the end, by means of nine evidences it is concluded that the exclusive mercy and favor of God are bestowed to human being so that out of them the real human creatures are selected to be mortal within the entity of God and everlasting upon His existence, and has made them special through their own knowledge and has adorned them with chastity so that the whole human creatures should abide by their province so as to get connected to His own province. These authentic connected human beings who own provincial attraction and are able to capture and witness the actions and inner heart of others hold the status of divine directors and Imams. The holy Koran introduces them with the adjectives including grace given, pious, truthful, martyrs, purified, chieftain, and the most loyal, and as long as human being exists, only these individuals are the human leaders toward the divine province. As a result, the chaste Imam is a comprehensive emblem of the right path.

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.

Let $n_1, n_2,\ldots,n_k$ be integers and $V_1, V_2,\ldots,V_k$ be disjoint vertex sets with $|V_i|=n_i$ for each $i= 1, 2,\ldots,k$. A $k$-partite $k$-uniform hypergraph on vertex classes $V_1, V_2,\ldots,V_k$ is defined to be the $k$-uniform hypergraph whose edge set consists of the $k$-element subsets $S$ of $V_1 \cup V_2 \cup \cdots \cup V_k$ such that $|S\cap V_i|=1$ for all $i= 1, 2,\ldots,k$. We say that it is balanced if $n_1=n_2=\cdots=n_k$. In this paper, we give a distance spectral radius condition to guarantee the existence of PERFECT MATCHING in $k$-partite $k$-uniform hypergraphs, this result generalize the result of Zhang and Lin  [PERFECT MATCHING and distance spectral radius in graphs and bipartite graphs, Discrete Appl. Math., 304 (2021) 315-322].

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.

In this article, we complement the study of Pan and Li by computing the first five minimum values of the symmetric division degree (SDD) index attained by bicyclic graphs that have a PERFECT MATCHING. One of our main contributions is identifying the graphs that attain the bounds. Further, we compute the upper bound of the SDD index for bicyclic graphs with a maximum degree of four, which admits a PERFECT MATCHING and prove the bound is also tight by identifying the graphs that attain it.

We find recursive formulae for the number of PERFECT MATCHINGs in a graph G by splitting G into subgraphs H and Q. We use these formulas to count PERFECT MATCHING of P hypercube Qn. We also apply our formulas to prove that the number of PERFECT MATCHING in an edge-transitive graph is Pm(G)=(2q/p)Pm(G\{u, n})where Pm(G) denotes the number of PERFECT MATCHINGs in G, G\{ u, n} is the graph constructed from G by deleting edges with an end vertex in {u, n} and unÎE(G).

In this paper, we obtain the upper and lower bounds on the eccentricity connectivity index of unicyclic graphs with PERFECT MATCHINGs. Also we give some lower bounds on the eccentric connectivity index of unicyclic graphs with given MATCHING numbers.

‎A PERFECT star packing in a given graph $G$ can be defined as a spanning subgraph of $G$‎, ‎wherein each component is isomorphic to the star graph $K_{1,3}$‎. ‎A PERFECT star packing of a fullerene graph $G$ is of type $P0$ if all the centers of stars lie on hexagons of $G$‎. ‎Many fullerene graphs arise from smaller fullerene graphs by applying some transformations‎. ‎In this paper‎, ‎we introduce two transformations for fullerene graphs that have the PERFECT star packing of type $P0$ and examine some characteristics of the graphs obtained from this transformation‎.