Graphs with the second signless Laplacian eigenvalue ≤ 4

We discuss the question of classifying the connected simple graphs H for which the second largest eigenvalue of the signless Laplacian Q(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.

Kolmogorov Basic Graphs and Their Application in Network Complexity Analysis

Throughout the years, measuring the complexity of networks and graphs has been of great interest to scientists. The Kolmogorov complexity is known as one of the most important tools to measure the complexity of an object. We formalized a method to calculate an upper bound for the Kolmogorov complexity of graphs and networks. Firstly, the most simple graphs possible, those with O(1) Kolmogorov complexity, were identified. These graphs were then used to develop a method to estimate the complexity of a given graph. The proposed method utilizes the simple structures within a graph to capture its non-randomness. This method is able to capture features that make a network closer to the more non-random end of the spectrum. The resulting algorithm takes a graph as an input and outputs an upper bound to its Kolmogorov complexity. This could be applicable in, for example evaluating the performances of graph compression methods.

Beyond Topological Persistence: Starting from Networks

Persistent homology enables fast and computable comparison of topological objects. We give some instances of a recent extension of the theory of persistence, guaranteeing robustness and computability for relevant data types, like simple graphs and digraphs. We focus on categorical persistence functions that allow us to study in full generality strong kinds of connectedness—clique communities, k-vertex, and k-edge connectedness—directly on simple graphs and strong connectedness in digraphs.

Three Topological Indices of Two New Variants of Graph Products

Graph operations play an important role to constructing complex network structures from simple graphs, and these complex networks play vital roles in different fields such as computer science, chemistry, and social sciences. Computation of topological indices of these complex network structures via graph operation is an important task. In this study, we defined two new variants of graph products, namely, corona join and subdivision vertex join products and investigated exact expressions of the first and second Zagreb indices and first reformulated Zagreb index for these new products.

A Generalization of Stiebitz-Type Results on Graph Decomposition

In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization of several results by various researchers. Let $G$ be a multigraph in which no quadrilaterals share edges with triangles and other quadrilaterals and let $\mu_G(v)=\max\{\mu_G(u,v):u\in V(G)\setminus\{v\}\}$, where $\mu_G(u,v)$ is the number of edges joining $u$ and $v$ in $G$. We show that for any two functions $a,b:V(G)\rightarrow\mathbb{N}\setminus\{0,1\}$, if $d_G(v)\ge a(v)+b(v)+2\mu_G(v)-3$ for each $v\in V(G)$, then there is a partition $(X,Y)$ of $V(G)$ such that $d_X(x)\geq a(x)$ for each $x\in X$ and $d_Y(y)\geq b(y)$ for each $y\in Y$. This extends the related results due to Diwan, Liu–Xu and Ma–Yang on simple graphs to the multigraph setting.

The Lay of the Land

This chapter explores the database of conspiracy claims in order to lay the groundwork for the analysis in future chapters. It begins by describing how the database was created and presenting an overview of its contents, including simple graphs showing where conspiracies (supposedly) take place, the nationalities of the accusers, and the identities of the perpetrators. To give a sense of the stories the claims tell, it homes in on three narrative elements of conspiracy—goals, actions, and logics—and provides examples. Finally, it breaks down the data according to combinations of accusers and perpetrators. Perusing the conspiracy claims reveals how they emerge from quotidian political realities, but in a milieu pervaded by intrigue, insecurity, and uncertainty. These claims tell a story—actually, several—about politics, but revolve around two questions: Who is doing what to whom and what do the perpetrators hope to accomplish?