We study pretty good quantum state transfer (ie, state transfer that becomes arbitrarily close to perfect) between vertices of graphs with an involution in the presence of an energy potential. In particular, we show that if a graph has an involution that fixes at least one vertex or at least one edge, then there exists a choice of potential on the vertex set of the graph for which we get pretty good state transfer between symmetric vertices of the graph. We show further that in many cases, the potential can be chosen so that it is only non-zero at the vertices between which we want pretty good state transfer. As a special case of this, we show that such a potential can be chosen on the endpoints of a path to induce pretty good state transfer in paths of any length. This is in contrast to the result of [6], in which the authors show that there cannot be perfect state transfer in paths of length 4 or more, no matter what potential is chosen.

We classify all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat. We use the definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Anal., 2009. A graph is Ricci-flat, if it has vanishing Ricci curvature on all edges. We show, that the only Ricci-flat cubic graphs with girth at least 5 are the Petersen graph, the Triplex and the dodecahedral graph. This will correct the classification in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex.

We associate to a finite digraph D a lattice polytope PD whose vertices are the rows of the Laplacian matrix of D. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of PD equals the complexity of D, and PD contains the origin in its relative interior if and only if D is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, the h∗-polynomial, and the integer decomposition property of PD in these cases. We extend Braun and Meyer’s study of cycles by considering cycle digraphs. In this setting, we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.

"A sequence $S$ is potentially $K_4-e$ graphical if it has a realization containing a $K_4-e$ as a subgraph. Let $\sigma(K_4-e,n)$ denote the smallest degree sum such that every $n$ -term graphical sequence $S$ with $\sigma(S)\geq\sigma(K_4-e,n)$ is potentially $K_4-e$ graphical. Gould, Jacobson, Lehel raised the problem of determining the value of $\sigma(K_4-e,n)$ . In this paper, we prove that $\sigma(K_4-e,n)=2[(3n-1)/2]$ for $n\geq7$ and $n=4,5$ , and $\sigma(K_4-e,6)=20$ .''

Let Q be an acyclic quiver and s be a sequence with elements in the vertex set Q_0. We describe an induced sequence of simple (backward) tilting in the bounded derived category D(Q), starting from the standard heart H_Q = modkQ and ending at another heart H_s in D(Q). Then we show that s is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to HQ[−1]; it is maximal if and only if H_s = H_Q[−1]. This provides a categorical way to understand green mutations. Further, fix a Coxeter element c in the Coxeter group W_Q of Q, which is admissible with respect to the orientation of Q. We prove that the sequence \tilde{w} induced by a c-sortable word w is a green mutation sequence. As a consequence, we obtain a bijection between c-sortable words and finite torsion classes in H_Q. As byproducts, the interpretations of inversions, descents and cover reflections of a c-sortable word w are given in terms of the combinatorics of green mutations.