When considering the total number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to some other graphical indices in applications. Along this line, it is interesting to study that over some types of trees with a given order, which trees minimize or maximize this number...

We study monogamy and polygamy inequalities of unified entanglement in multipartite quantum systems. first derive the inequality unified-(q, s) for multi-qubit states under arbitrary bipartition, then obtain $$\alpha $$ th ( $$0\le \alpha \le \frac{r}{2}, r\ge \sqrt{2}$$ ) power formation tripartite their generalizations states. also generalize bipartition. Moreover, we investigate $$\beta \ge ...

Given a tree T on n vertices and a set P of n points in the plane in general position, it is known that T can be straight line embedded in P without crossings. The problem becomes more diicult if T is rooted and we want to root it at any particular point of P. The problem in this form was posed by Perles and partially solved by Pach and Torosick 5]. A complete solution was found by Ikebe et al....

The tree representation as a model for organismal evolution has been in use since before Darwin. However, with the recent unprecedented access to biomolecular data, it has been discovered that, especially in the microbial world, individual genes making up the genome of an organism give rise to different and sometimes conflicting evolutionary tree topologies. This discovery calls into question t...

We extend the work of Letchford (2000) by introducing a new class of valid inequalities for the traveling salesman problem, called the generalized domino-parity (GDP) constraints. Just as Letchford’s domino-parity constraints generalize comb inequalities, GDP constraints generalize the most well-known multiple-handle constraints, including clique-tree, bipartition, path, and star inequalities. ...

We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms G and are either locally (G, s)– arc transitive for s ≥ 2 or G–locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of G. Given a normal subgroup N which is intransitive on both parts of the bipartition, we show that taking quotients with r...

A medial cuneiform exhibiting complete bipartition was discovered at the Early Pleistocene site of Dmanisi, Georgia. The specimen is the oldest known instance of this anatomical variant in the hominin fossil record. Here we compare developmental variation of the medial cuneiform in fossil hominins, extant humans and great apes, and discuss potential implications of bipartition for hominin foot ...

A tree is called a k-tree if its maximum degree is at most k. We prove the following theorem. Let k ⩾ 2 be an integer, and G be a connected bipartite graph with bipartition (A,B) such that |A| ⩽ |B| ⩽ (k − 1)|A| + 1. If σk(G) ⩾ |B|, then G has a spanning k-tree, where σk(G) denotes the minimum degree sum of k independent vertices of G. Moreover, the condition on σk(G) is sharp. It was shown by ...