Simulation of defects, flexibility and rupture in biopolymer networks

We use a coarse grained polymer model and a simple graph representation to introduce defects into a biopolymer network, then cause them to rupture.

Computing some role assignments of Cartesian Product of Graphs

In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. As a simple graph models a social network role assignment rises to the decision problem called r -Role Assignment whether it exists such an assignment of r distinct roles to the vertices of the graph. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is one of the most studied operation on graphs and has numerous applications in diverse areas, such as Mathematics, Computer Science, Chemistry and Biology. In this paper, we determine the computational complexity of r -Role Assignment restricted to Cartesian product of graphs, for r = 2,3. In fact, we show that the Cartesian product of graphs is always 2-role assignable, however the problem of 3-Role Assignment is still NP-complete for this class.

NOTE ON SUPER \((a,1)\)–\(P_3\)–ANTIMAGIC TOTAL LABELING OF STAR \(S_n\)

Let \(G=(V, E)\) be a simple graph and \(H\) be a subgraph of \(G\). Then \(G\) admits an \(H\)-covering, if every edge in \(E(G)\) belongs to at least one subgraph of \(G\) that is isomorphic to \(H\). An \((a,d)-H\)-antimagic total labeling of \(G\) is bijection \(f:V(G)\cup E(G)\rightarrow \{1, 2, 3,\dots, |V(G)| + |E(G)|\}\) such that for all subgraphs \(H'\) of \(G\) isomorphic to \(H\), the \(H'\) weights \(w(H') =\sum_{v\in V(H')} f (v) + \sum_{e\in E(H')} f (e)\) constitute an arithmetic progression \(\{a, a + d, a + 2d, \dots , a + (n- 1)d\}\), where \(a\) and \(d\) are positive integers and \(n\) is the number of subgraphs of \(G\) isomorphic to \(H\). The labeling \(f\) is called a super \((a, d)-H\)-antimagic total labeling if \(f(V(G))=\{1, 2, 3,\dots, |V(G)|\}.\) In [5], David Laurence and Kathiresan posed a problem that characterizes the super \( (a, 1)-P_{3}\)-antimagic total labeling of Star \(S_{n},\) where \(n=6,7,8,9.\) In this paper, we completely solved this problem.

Improved Selection of Rare Reactions in Template-Based Retrosynthesis Predictions

Identifying synthetic routes for molecules of interest is a crucial step when discovering new drugs or materials. To find synthetic routes, we can use computer-assisted synthesis planning using expansion policy networks trained on reaction templates extracted from patents and the literature. However, experience has shown that these networks are biased towards frequently reported reactions. This study shows that changing the molecular representation from an extended-connectivity fingerprint to a simple graph representation can increase the accuracy for templates used less than five times by 5.0- 8.5% points. We also illustrate that a simple oversampling of the training set yielded a top-1 accuracy increase in the 17-20% point range for templates used five times or less.

Edge-vertex domination in trees

Let [Formula: see text] be a finite simple graph. A vertex [Formula: see text] is edge-vertex dominated by an edge [Formula: see text] if [Formula: see text] is incident with [Formula: see text] or [Formula: see text] is incident with a vertex adjacent to [Formula: see text]. An edge-vertex dominating set of [Formula: see text] is a subset [Formula: see text] such that every vertex of [Formula: see text] is edge-vertex dominated by an edge of [Formula: see text]. The edge-vertex domination number [Formula: see text] is the minimum cardinality of an edge-vertex dominating set of [Formula: see text]. In this paper, we prove that [Formula: see text] for every tree [Formula: see text] of order [Formula: see text] with [Formula: see text] leaves, and we characterize the trees attaining each of the bounds.

3-Total Edge Product Cordial Labeling for Stellation of Square Grid Graph

Let G be a simple graph with vertex set V G and edge set E G . An edge labeling δ : E G ⟶ 0,1 , … , p − 1 , where p is an integer, 1 ≤ p ≤ E G , induces a vertex labeling δ ∗ : V H ⟶ 0,1 , … , p − 1 defined by δ ∗ v = δ e 1 δ e 2 ⋅ δ e n mod p , where e 1 , e 2 , … , e n are edges incident to v . The labeling δ is said to be p -total edge product cordial (TEPC) labeling of G if e δ i + v δ ∗ i − e δ j + v δ ∗ j ≤ 1 for every i , j , 0 ≤ i ≤ j ≤ p − 1 , where e δ i and v δ ∗ i are numbers of edges and vertices labeled with integer i , respectively. In this paper, we have proved that the stellation of square grid graph admits a 3-total edge product cordial labeling.

AdjMix: simplifying and attending graph convolutional networks

Simple graph convolution (SGC) achieves competitive classification accuracy to graph convolutional networks (GCNs) in various tasks while being computationally more efficient and fitting fewer parameters. However, the width of SGC is narrow due to the over-smoothing of SGC with higher power, which limits the learning ability of graph representations. Here, we propose AdjMix, a simple and attentional graph convolutional model, that is scalable to wider structure and captures more nodes features information, by simultaneously mixing the adjacency matrices of different powers. We point out that the key factor of over-smoothing is the mismatched weights of adjacency matrices, and design AdjMix to address the over-smoothing of SGC and GCNs by adjusting the weights to matching values. Experiments on citation networks including Pubmed, Citeseer, and Cora show that our AdjMix improves over SGC by 2.4%, 2.2%, and 3.2%, respectively, while achieving same performance in terms of parameters and complexity, and obtains better performance in terms of classification accuracy, parameters, and complexity, compared to other baselines.

Mathematical model and algorithm for calculating the cycles of the cells of the graph map

Выделенные свойства циклов DFS-базиса блока карты простого графа позволили составить математическую модель вычисления циклов ячеек карты графа. По данной модели предложен практический алгоритм вычисления циклов ячеек карты графа. Алгоритм имеет квадратическую сложность относительно числа вершин в графе. The selected properties of the cycles of the DFS-basis block of a simple graph map allowed us to create a mathematical model for calculating the cycles of the cells of the graph map. According to this model, a practical algorithm for calculating the cycles of the graph map cells is proposed. The algorithm has a quadratic complexity relative to the number of vertices in the graph.

On k-rainbow domination in middle graphs

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.