We describe a new approach for cluster-based drawing of large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSP-type decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounded aspect ratios. In addition, the tree depth is O(log n), and its construction takes O(n log n) t...

We describe a new approach for cluster-based drawing of large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSP-type decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounded aspect ratios. In addition, the tree depth is O(log n), and its construction takes O(n log n) t...

let $p$ be a prime number and $n$ be a positive integer. the graph $g_p(n)$ is a graph with vertex set $[n]={1,2,ldots ,n}$, in which there is an arc from $u$ to $v$ if and only if $uneq v$ and $pnmid u+v$. in this paper it is shown that $g_p(n)$ is a perfect graph. in addition, an explicit formula for the chromatic number of such graph is given.

• We adapt a traditional non-learnable GED algorithm to the novel paradigm of geometric deep learning. Triplet network for learning graph distances by means neural networks. Learning distance in domain without an embedding stage. Graph-based keyword spotting application with state-of-the-art performance. The emergence as framework deal graph-based representations has faded away approaches favor...

This paper introduces QuanTree v1.1 and QuanLin v1.1, two Java applications available for free. (Source code included in the distribution.) Each application compiles a different type of input quantum evolution operator. The applications output a quantum circuit that is approximately equal to the input evolution operator. QuanTree compiles an input evolution operator whose Hamiltonian is proport...

For any k ∈ N, the k−distance graph DG has the same vertex set of G, and two vertices of DG are adjacent if they are exactly distance k apart in the original graph G. In this paper, we consider the connectivity of DG and state the conditions for graph G and integer k such that the graph DG is connected.

A graph G is called P3-dominated (P3D) if it satisfies J(x, y) ∪ J ′(x, y) 6= ∅ for every pair (x, y) of vertices at distance 2, where J(x, y) = {u|u ∈ N(x) ∩N(y), N [u] ⊆ N [x]∪N [y]} and J ′(x, y) = {u|u ∈ N(x)∩N(y)| if v ∈ N(u)\(N [x]∪N [y]), then (N(u)∪ N(x) ∪ N(y))\{x, y, v} ⊆ N(v)} for x, y ∈ V (G) at distance 2}. For a noncomplete graph G, the number NC is defined as NC = min{|N(x) ∪ N(y...

1. Suppose you are given a graph whose edge lengths are all integers in the range from 0 to B. Suppose also that you are given the all-pairs distance matrix for this graph (it can be constructed by a variant of Seidel’s distance algorithm). Prove that you can identify the (successor matrix representation of the) shortest paths in O(BMM(n) log n) time, where MM(n) is the time to multiply n × n m...