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let g be a graph with p vertices and q edges and a = {0, 1, 2, . . . , [q/2]}. a vertex labeling f : v (g) → a induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. for a ∈ a, let vf (a) be the number of vertices v with f(v) = a. a graph g is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in a, |vf (a) − vf (b)| ≤ 1 and the in...

The success of Google’s Pregel framework in distributed graph processing has inspired a surging interest in developing Pregel-like platforms featuring a user-friendly “think like a vertex” programming model. Existing Pregel-like systems support a fault tolerance mechanism called checkpointing, which periodically saves computation states as checkpoints to HDFS, so that when a failure happens, co...

For a graph G = (V,E) with a binary vertex coloring f : V (G)→ Z2, let vf (i) = |f−1(i)|. We say f is friendly if |vf (1) − vf (0)| ≤ 1, i.e., the number of vertices labeled 1 is the same or almost the same as the number of vertices labeled 0. The coloring f induces an edge labeling f ∗ : E(G) → Z2 defined by f ∗(uv) = f(u) + f(v) (mod 2), for each uv ∈ E(G). Let ef (i) = |{uv ∈ E(G) : f ∗(uv) ...

In this paper, we propose a GPU-friendly shape interpolation method. In contrast with state-of-the-art interpolation algorithms, our method computes the trajectory of each vertex independently instead of solving large linear systems in every interpolation step. Given two poses being interpolated, we find trajectory parameters for each vertex by optimization with the consideration of the key pos...

In this paper, we present Vertexica, a graph analytics tools on top of a relational database, which is user friendly and yet highly e›cient. Instead of constraining programmers to SQL, Vertexica offers a popular vertex-centric query interface, which is more natural for analysts to express many graph queries. e programmers simply provide their vertex-compute functions and Vertexica takes care of...

Polygon, subdivision, and NURBS are three mainstream modeling techniques widely applied in commercial software packages. They require heavy manual operations, involve a lot of design variables leading to big data, high storage costs slow network transmissions. In this paper, we integrate the strengths boundary-based surface creation partial differential equation (PDE)-based geometric obtain fir...

‎let $g$ be a graph and $chi^{prime}_{aa}(g)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $g$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for $chi^{prime}_{aa}(gsquare h)$ for any two graphs $g$ and $h$‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of ...

in this paper, at first we mention to some results related to pi and vertex co-pi indices and then we introduce the edge versions of co-pi indices. then, we obtain some properties about these new indices.