Different kinds of decompositions of a square matrix have been studied in the literature by altering the conditions on the summands of core-nilpotent decomposition. Motivated by the observation that for an element from an associative ring the Drazin inverse and the core-nilpotent decomposition coexist, and many of among the techniques which are useful in the case of matrices fail in the case dealing with the elements from an associative ring, the theory of generalized inverse and the minus partial order are used to characterize and study the generalized core-nilpotent decomposition of an element from an associative ring.

The Drazin inverse is connected with the notion of index and core-nilpotent decomposition whenever it is discussed in the context of ring of matrices over complex field. In the absence of Drazin inverse for a given element from an arbitrary associative ring (not necessarily with unity), in this paper, the notion of right (left) core-nilpotent decomposition has been introduced and established its relations with right (left) [Formula: see text]-regular property. In fact, the class of such decomposition has been characterized. In case of regular ring, observed that an element is right (left) [Formula: see text]-regular if and only if it has a right (left) core-nilpotent decomposition. In the process, several properties of sharp order in an associative ring are studied and with the help of the same, new characterizations of Drazin inverse over an associative ring are obtained and the relation between core-nilpotent decomposition and the Drazin inverse is obtained.

In this article, a relation on the class of real rectangular matrices based on the involution of secondary-transpose, called s-order, and a $$\mathcal {G}$$ -based relation on the same class, called $$\dag _s$$ -order using the s-g inverse, are introduced. In Sect. 3, a new necessary and sufficient condition for the existence of s-g inverse with reference to s-symmetric projectors is provided. In Sect. 4, the properties of the new relations defined are studied and noted that $$\le ^s$$ and $$\le ^{\dag _s}$$ are partial orders on the set of all matrices having s-g inverse. Motivated by the earlier works on star order, in Sect. 5, the column space of factors of given matrices with reference to the relations considered are characterized. Proving that there is a one-one correspondence between invariant subspace of $$AA^s$$ having s-symmetric projectors and the matrices B such that $$B\le ^{\dag _s}A$$ , rank 1 factors are characterized. Also, in Sect. 6, a new decomposition which establishes a relationship between s-g inverse of the given matrix and s-g inverses of its components is discussed.

In this article, some interesting applications of generalized inverses in the graph theory are revisited. Interesting properties of generalized inverses are employed to make the proof of several known results simpler, and several techniques such as bordering method and inverse complemented matrix methods are used to obtain simple expressions for the Moore-Penrose inverse of incidence matrix and Laplacian matrix. Some interesting and simpler expressions are obtained in some special cases such as tree graph, complete graph and complete bipartite graph.

The spectrum of a graph G is the set of eigenvalues of the adjacency matrix of G. The nullity $$\eta (G)$$ , which is the algebraic multiplicity of the number zero in the spectrum of G, is a graph spectrum-based invariant. In the context of the H $$\overset{..}{\text {u}}$$ ckel Molecular Orbital theory, the nullity of a molecular graph is used to determine the stability of unsaturated conjugated hydrocarbons. In this paper, we present a survey of significant results on nullity, such as bounds, transformations preserving nullity, nullity of line graphs, nullity versus energy and graphs with high nullity, along with some new results.

Stress is a centrality measure determined by the shortest paths passing through the given vertex. Noting that adjacency matrix playing an important role in finding the distance and the number of shortest paths between given pair of vertices, an interesting expression and also an algorithm are presented to find stress using adjacency matrix. The results and algorithm are suitably adopted to obtain betweenness centrality measure. Further results are extended to the cases of Cartesian product GH of graphs, corona graph G & DEG;H and their special cases.

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in the spectrum of G. A unified approach is presented for the characterization of graphs of order n with η(G) = n−4. All known results on trees, unicyclic graphs, bicyclic graphs, graphs with minimum degree 1, and r-partite graphs, for which η(G) = n−4 are shown to be corollaries of a theorem of Chang, Huang and Yeh that characterizes all graphs with nullity n − 4.

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

The reverse order law for outer inverses and the Moore-Penrose inverse is discussed in the context of associative rings. A class of pairs of outer inverses that satisfy reverse order law is determined. The notions of left-star and right-star orders have been extended to the case of arbitrary associative rings with involution and many of their interesting properties are explored. The distinct behavior of projectors in association with the star, right-star, and left-star partial orders led to several equivalent conditions for the reverse order law for the Moore-Penrose inverse.

Let G = (V, E) be a connected graph. The stress is a vertex centrality measure which assigns a real number st(v), called the stress of v, to each vertex of G. It is used in the study of biological and social networks for ranking the vertices and identifying important vertices in the network. In this paper, we determine the stress of wheel related graphs such as gear graph, helm graph, friendship graph, flower graph and sunflower graph.

Stress is an important centrality measure of graphs applicableto the study of social and biological networks. We study the stress of paths, cycles, fans andwheels. We determine the stress of a cut vertex of a graph G, when G has at most two cutvertices. We have also identified the graphs with minimum stress and maximum stress in thefamily of all trees of order $n$ and in the family of all complete bipartite graphs of order n.

In this paper, the concept of "Inverse Complemented Matrix Method", introduced by Eagambaram (2018), has been reestablished with the help of minus partial order and several new properties of complementary matrices and the inverse of complemented matrix are discovered. Class of generalized inverses and outer inverses of given matrix are characterized by identifying appropriate inverse complement. Further, in continuation, we provide a condition equivalent to the regularity condition for a matrix to have unique shorted matrix in terms of inverse complemented matrix. Also, an expression for shorted matrix in terms of inverse complemented matrix is given.

The 110Pd(n,2n)109Pd, 102Pd(n,2n)101Pd, 105Pd(n,p)105Rh and 106Pd(n,p)106mRh reaction cross sections have been measured with respect to the 197Au(n,2n)196Au monitor reaction at the neutron energy of 14.54 ± 0.24 MeV by using the method of activation and off-line γ-ray spectrometry. The mono-energetic neutron beam was generated from the D–T reaction. The uncertainties in the various basic nuclear parameters in the measured reactions and their correlations were estimated by using covariance analysis. The present data were compared with the EXFOR based literature data, evaluated data of various libraries available in national nuclear data center and with the calculated values from TALYS-1.9 code.

The cross sections have been estimated for the Nuclear reactions Mo-92(n,a)Zr-89 and Mo-97(n,p)Nb-97 produced in Purnima neutron generator at neutron energy of 13.52 +/- 0.0045 MeV using activation analysis and off-line gamma-ray spectrometric techniques. Al-27(n,alpha)Na-24 has been used as a monitor reaction. The covariance analysis for these cross sections has been carried out by taking into consideration of partial uncertainties of different attributes and correlations between the attributes. The cross section values of the present study have been compared with EXFOR, ENDF data of various libraries and theoretical data of TALYS-1.8 code.

The 115 In(n,2n) 114m In and 197 Au(n,2n) 196 Au reaction cross sections have been measured relative to the 27 Al(n,α) 24 Na monitor reaction at the two different incident neutron energies of 13.520 ± 0.005 MeV and 14.54 ± 0.24 MeV. The neutrons from the D-T fusion reaction at the Purnima neutron generator were used for activation followed by off-line γ-ray spectrometry. The uncertainty propagation and correlation of measured reaction cross sections were estimated using covariance analysis through considering the partial uncertainties in different attributes. The measured reaction cross sections from the present work have been compared with the literature data from EXFOR compilation, ENDF data of various libraries and theoretically calculated values from the TALYS-1.9 code.

The (n,2n) reaction cross sections of cerium isotopes have been experimentally measured at the neutron energy of 13.50 ± 0.15 MeV by using the method of activation and off-line γ-ray spectroscopy. The neutron energy was obtained from the D-T reaction using the Purnima Neutron Generator at BARC. The natCe sample was irradiated along with the gold and indium monitor foils. The cross sections of 140Ce(n,2n)139Ce and 142Ce(n,2n)141Ce reactions were measured by using both the gold and indium as a monitor and then collapsed to get the best values of reactions cross sections. The uncertainties in the measured cross sections of 140Ce(n,2n)139Ce and 142Ce(n,2n)141Ce reactions were determined using covariance analysis by considering uncertainties in various attributes. The measured values from the present work were compared with the literature data based on EXFOR compilation, evaluated data from different libraries and theoretically calculated values from TALYS-1.95 and EMPIRE-3.2.

The 115In(n,2n)114mIn and 197Au(n,2n)196Au reaction cross sections have been measured relative to the 27Al(n,α)24Na monitor reaction at the two different incident neutron energies of 13.520 ± 0.005 MeV and 14.54 ± 0.24 MeV. The neutrons from the D-T fusion reaction at the Purnima neutron generator were used for activation followed by off-line γ-ray spectrometry. The uncertainty propagation and correlation of measured reaction cross sections were estimated using covariance analysis through considering the partial uncertainties in different attributes. The measured reaction cross sections from the present work have been compared with the literature data from EXFOR compilation, ENDF data of various libraries and theoretically calculated values from the TALYS-1.9 code.

The cross sections for the 93 Nb(n,2n) 92m Nb, 93 Nb(n,α) 90m Y and the 92 Mo(n,p) 92m Nb reactions have been measured with respect to the 197 Au(n,2n) 196 Au monitor reaction at the incident neutron energy of 14.78 ± 0.19 MeV by employing methods of activation and off-line γ-ray spectrometry. The covariance analysis was carried out by taking into consideration of partial uncertainties in different attributes and correlation among the attributes. The present data have been compared with the literature data available in EXFOR, evaluated data of different libraries and theoretical values based on TALYS-1.8 code.