A generalization of heterochromatic graphs

In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose $f$-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is $f$-chromatic if each color $c$ appears on at most $f(c)$ edges. We also present a necessary and sufficient condition for edge-colored graphs to have an $f$-chromatic spanning forest with exactly $m$ components. Moreover, using this criterion, we show that a $g$-chromatic graph $G$ of order $n$ with $|E(G)|>\binom{n-m}{2}$ has an $f$-chromatic spanning forest with exactly $m$ ($1 \le m \le n-1$) components if $g(c) \le \frac{|E(G)|}{n-m}f(c)$ for any color $c$.Comment: 14 pages, 4 figure

Sibling Rivalry among Paralogs Promotes Evolution of the Human Brain

Geneticists have long sought to identify the genetic changes that made us human, but pinpointing the functionally relevant changes has been challenging. Two papers in this issue suggest that partial duplication of SRGAP2, producing an incomplete protein that antagonizes the original, contributed to human brain evolution

Sign patterns for chemical reaction networks

Most differential equations found in chemical reaction networks (CRNs) have the form $dx/dt=f(x)= Sv(x)$, where $x$ lies in the nonnegative orthant, where $S$ is a real matrix (the stoichiometric matrix) and $v$ is a column vector consisting of real-valued functions having a special relationship to $S$. Our main interest will be in the Jacobian matrix, $f'(x)$, of $f(x)$, in particular in whether or not each entry $f'(x)_{ij}$ has the same sign for all $x$ in the orthant, i.e., the Jacobian respects a sign pattern. In other words species $x_j$ always acts on species $x_i$ in an inhibitory way or its action is always excitatory. In Helton, Klep, Gomez we gave necessary and sufficient conditions on the species-reaction graph naturally associated to $S$ which guarantee that the Jacobian of the associated CRN has a sign pattern. In this paper, given $S$ we give a construction which adds certain rows and columns to $S$, thereby producing a stoichiometric matrix $\widehat S$ corresponding to a new CRN with some added species and reactions. The Jacobian for this CRN based on $\hat S$ has a sign pattern. The equilibria for the $S$ and the $\hat S$ based CRN are in exact one to one correspondence with each equilibrium $e$ for the original CRN gotten from an equilibrium $\hat e$ for the new CRN by removing its added species. In our construction of a new CRN we are allowed to choose rate constants for the added reactions and if we choose them large enough the equilibrium $\hat e$ is locally asymptotically stable if and only if the equilibrium $e$ is locally asymptotically stable. Further properties of the construction are shown, such as those pertaining to conserved quantities and to how the deficiencies of the two CRNs compare.Comment: 23 page

The interval structure of (0,1)-matrices

Let A be an n × n (0, ∗)-matrix, so each entry is 0 or ∗. An A-interval matrix is a (0, 1)-matrix obtained from A by choosing some ∗’s so that in every interval of consecutive ∗’s, in a row or column of A, exactly one ∗ is chosen and replaced with a 1, and every other ∗ is replaced with a 0. We consider the existence questions for A-interval matrices, both in general, and for specific classes of such A defined by permutation matrices. Moreover, we discuss uniqueness and the number of A-permutation matrices, as well as properties of an associated graph

Alternating Sign Matrices -- Extensions, Konig-properties, and Primary Sum-Sequences

This paper is concerned with properties of permutation matrices and alternating sign matrices (ASMs). An ASM is a square (0,±1)-matrix such that, ignoring 0’s, the 1’s and −1’s in each row and column alternate, beginning and ending with a 1. We study extensions of permutation matrices into ASMs by changing some zeros to +1 or −1. Furthermore, several properties concerning the term rank and line covering of ASMs are shown. An ASM A is determined by a sum-matrix Σ(A) whose entries are the sums of the entries of its leading submatrices (so determined by the entries of A). We show that those sums corresponding to the nonzero entries of a permutation matrix determine all the entries of the sum-matrix and investigate some of the properties of the resulting sequence of numbers. Finally, we investigate the lattice-properties of the set of ASMs (of order n), where the partial order comes from the Bruhat order for permutation matrices

Alternating Sign Matrices, Related (0, 1)-Matrices, and the Smith Normal Form

We investigate the Smith Normal Form (SNF) of alternating sign matrices (ASMs) and related matrices of 0’s and 1’s ((0, 1)-matrices). We identify certain classes of ASMs and (0, 1)-matrices whose SNFs are (0, 1)-matrices. We relate some of our work to various ranks, in particular, the (0, 1)-rank of a (0, 1)-matrix, that is, the bipartite partition number of a bipartite graph