In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of paths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.

In 1975, P. Erd\H os proposed the problem of determining the maximum number $f(n)$ of edges in a graph with $n$ vertices in which any two cycles are of different lengths. The sequence $(c_1,c_2,\cdots,c_n)$ is the cycle length distribution of a graph $G$ of order $n$， where $c_i$ is the number of cycles of length $i$ in $G$. Let $f(a_1,a_2,\cdots,$ $a_n)$ denote the maximum possible number of edges in a graph which satisfies $c_i\leq a_i$ where $a_i$ is a nonnegative integer. In 1991, Shi posed the problem of determining $f(a_1,a_2,\cdots,a_n)，$ which extended the problem due to Erd\H os, it is clear that $f(n)=f(1,1,\cdots,1)$. Let $g(n,m)=f(a_1,a_2,\cdots,a_n),$ $a_i=1$ for all $i/m$ be integer, $a_i=0$ for all $i/m$ be not integer. It is clear that $f(n)=g(n,1)$. We prove that $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{40}{99}},$ which is better than the previous bounds $\sqrt 2$ (Shi, 1988), $\sqrt {2 + \frac{7654}{19071}}$ (Lai, 2017). We show that $\liminf_{n \rightarrow \infty} {g(n,m)-n\over \sqrt \frac{n}{m}} > \sqrt {2.444},$ for all even integers $m$. We make the following conjecture: $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} > \sqrt {2.444}.$

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We study a certain discrete differentiation of piecewise-constant functions on the adjoint of the braid hyperplane arrangement, defined by taking finite-differences across hyperplanes. In terms of Aguiar-Mahajan's Lie theory of hyperplane arrangements, we show that this structure is equivalent to the action of Lie elements on faces. We use layered binary trees to encode flags of adjoint arrangement faces, allowing for the representation of certain Lie elements by antisymmetrized layered binary forests. This is dual to the well-known use of (delayered) binary trees to represent Lie elements of the braid arrangement. The discrete derivative then induces an action of layered binary forests on piecewise-constant functions, which we call the forest derivative. Our main result states that forest derivatives of functions factorize as external products of functions precisely if one restricts to functions which satisfy the Steinmann relations, which are certain four-term linear relations appearing in the foundations of axiomatic quantum field theory. We also show that the forest derivative satisfies the Lie properties of antisymmetry the Jacobi identity. It follows from these Lie properties, and also crucially factorization, that functions which satisfy the Steinmann relations form a left comodule of the Lie cooperad, with the coaction given by the forest derivative. Dually, this endows the adjoint braid arrangement modulo the Steinmann relations with the structure of a Lie algebra internal to the category of vector species. This work is a first step towards describing new connections between Hopf theory in species and quantum field theory.

In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph with $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+\frac{107}{3}t+\frac{7}{3}$$ for $t=1260r+169 \,\ (r\geq 1)$ and $n \geq \frac{2119}{4}t^{2}+87978t+\frac{15957}{4}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{7654}{19071}},$ which is better than the previous bounds $\sqrt 2$ (Shi, 1988), $\sqrt {2.4}$ (Lai, 2003). The conjecture $\lim_{n \rightarrow \infty} {f(n)-n\over \sqrt n}=\sqrt {2.4}$ is not true.