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Take three integers $$m\ge 0,\,k\ge 1$$ , and $$n\ge 2$$ . Let $$a\ (\not \equiv 0)$$ be a holomorphic function in a domain $$D$$ of $$\mathbb {C}$$ such that multiplicities of zeros of $$a$$ are at most $$m$$ and divisible by $$n+1$$ . In this paper, we mainly obtain the following normality criterion: Let $${{{\fancyscript{F}}}}$$ be the family of meromorphic functions on $$D$$ such that multiplicities of zeros of each $$f\in {{\fancyscript{F}}}$$ are at least $$k+m$$ and such that multiplicities of poles of $$f$$ are at least $$m+1$$ . If each pair $$(f,g)$$ of $${{\fancyscript{F}}}$$ satisfies that $$f^{n}f^{(k)}$$ and $$g^{n}g^{(k)}$$ share $$a$$ (ignoring multiplicity), then $${{\fancyscript{F}}}$$ is normal. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Morgenstern type bivariate-generalized exponential distribution. This distribution is an extension of Morgenstern type bivariate exponential distribution (MTBED), and the marginal distributions are generalized exponential distribution. We study some properties of this bivariate distribution. Also, some distributional properties of concomitants of order statistics as well as record values for the MTBED are studied. Recurrence relations between moments of concomitants are also obtained. [ABSTRACT FROM AUTHOR]

The b-chromatic index $$\varphi '(G)$$ of a graph $$G$$ is the largest integer $$k$$ such that $$G$$ admits a proper $$k$$ -edge coloring in which every color class contains at least one edge incident to some edge in all the other color classes. The b-chromatic index of trees is determined and equals either to a natural upper bound $$m'(T)$$ or one less, where $$m'(T)$$ is connected with the number of edges of high degree. Some conditions are given for which graphs have the b-chromatic index strictly less than $$m'(G)$$ , and for which conditions it is exactly $$m'(G)$$ . In the last part of the paper, regular graphs are considered. It is proved that with four exceptions, the b-chromatic index of cubic graphs is $$5$$ . The exceptions are $$K_4$$ , $$K_{3,3}$$ , the prism over $$K_3$$ , and the cube $$Q_3$$ . [ABSTRACT FROM AUTHOR]

A connected graph $$G$$ is said to be factor-critical if $$G-v$$ has a perfect matching for every vertex $$v$$ of $$G$$ . Lovász proved that every factor-critical graph has an ear decomposition. In this paper, the ear decomposition of the factor-critical graphs $$G$$ satisfying that $$G-v$$ has a unique perfect matching for any vertex $$v$$ of $$G$$ with degree at least 3 is characterized. From this, the number of maximum matchings of factor-critical graphs with the special ear decomposition is obtained. [ABSTRACT FROM AUTHOR]

An $$m\times n\,\{0,1\}$$ -matrix $$A$$ is said to be totally convertible if there exists a matrix $$B$$ obtained from $$A$$ by changing some 1's in $$A$$ to $$-1$$ 's such that for any submatrix $$A^{\prime }$$ of $$A$$ of order $$m$$ , the corresponding submatrix $$B^{\prime }$$ of $$B$$ satisfies $$\mathrm{per}(xI-A^{\prime })=\det (xI-B^{\prime })$$ . In this paper, motivated by the well-known Pólya's problem, our object is to characterize those totally convertible matrices. Associate a matrix $$A$$ with a bipartite graph $$G^{*}_A$$ . We first prove that a square matrix $$A$$ is totally convertible if and only if $$G^{*}_A$$ is Pfaffian, and then we generalize this result to an $$m\times n$$ $$\{0,1\}$$ -matrix. Moreover, the characterization of a totally convertible matrix provides an equivalent condition to compute the permanental polynomial of a bipartite graph by the characteristic polynomial of the skew adjacency matrix of its orientation graph. As applications, we give some explicit expressions of the permanental polynomials of two totally convertible matrices by the technique of Pfaffian orientation. [ABSTRACT FROM AUTHOR]

In this paper, we present some results on the existence of random coincidence points of expansive type completely random operators. Some applications to random fixed point theorems and random equations are given. [ABSTRACT FROM AUTHOR]