Search Results (9)

In 2017, Nikiforov defined the $A_{\alpha }$-matrix of the graph G as $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right),0\le \alpha \le 1,$ which merges the diagonal degree matrix $D\left(G\right)$ and the adjacency matrix $A\left(G\right)$. In this paper, we characterize the graphs which attain the maximum $A_{\alpha }$-index among triangle-free non-bipartite graphs of order n for $1/2<\alpha <1$. Full article

The threshold protocol game is a graphical game that models the adoption of an idea or product through a population. There are two states players may take in the game, and the goal of the game is to motivate the state that begins in the minority to spread to every player. Here, the threshold protocol game is defined, and existence results are studied on infinite graphs. Many generalizations are proposed and applied. This work explores the impact of graph topology on the outcome of the threshold protocol game and consequently considers finite graphs. By exploiting the well-known topologies of complete and complete bipartite graphs, the outcome of the threshold protocol game can be fully characterized on these graphs. These characterizations are ideal, as they are given in terms of the game parameters. More generally, initial conditions in terms of game parameters that cause the preferred game outcome to occur are identified. It is shown that the necessary conditions differ between non-bipartite and bipartite graphs because non-bipartite graphs contain odd cycles while bipartite graphs do not. These results motivate the primary result of this work, which is an exhaustive list of achievable game outcomes on bipartite graphs. While possible outcomes are identified, it is noted that a complete characterization of when game outcomes occur is not possible on general bipartite graphs. Full article

In the paper, we introduce the signless Laplacian $ABC$-matrix $\tilde{\mathrm{Q}}(G)=\overline{D}(G)+\tilde{\mathrm{A}}(G)$, where $\overline{D}(G)$ is the diagonal matrix of $ABC$-degrees and $\tilde{\mathrm{A}}(G)$ is the $ABC$-matrix of $G.$ The eigenvalues of the matrix $\tilde{\mathrm{Q}}(G)$ are the signless Laplacian $ABC$-eigenvalues of G. We give some basic properties of the matrix $\tilde{\mathrm{Q}}(G)$, which includes relating independence number and clique number with signless Laplacian $ABC$-eigenvalues. For bipartite graphs, we show that the signless Laplacian $ABC$-spectrum and the Laplacian $ABC$-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian $ABC$-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian $ABC$-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix $\tilde{\mathrm{Q}}(G)-{\displaystyle \frac{tr(\tilde{\mathrm{Q}}(G))}{n}}I$, called the signless Laplacian $ABC$-energy of G. We obtain some upper and lower bounds for signless Laplacian $ABC$-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the $ABC$-energy with the signless Laplacian $ABC$-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian $ABC$-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same $ABC$-energy. Full article

Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum–optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows extracting high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo integration enables calculations on large finite systems. Finite-size scaling can then be used to determine the physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations. Full article

The vertex quadrangulation $QG$ of a 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. In a previous work [JOMC 59, 1551–1569 (2021)], the question was posed: does the spectrum of an arbitrary unweighted graph $QG$ include the full spectrum $\{3,{(-1)}^3\}$ of the tetrahedron graph (complete graph $K_4$)? Previously, many bipartite and nonbipartite graphs $QG$ with such a subspectrum have been found; for example, a nonbipartite variant of the graph $Q{K}_5$. Here, we present one of the variants of the nonbipartite vertex quadrangulation $QO$ of the octahedron graph O, which has eigenvalue $(-1)$ of multiplicity 2 in the spectrum, while the spectrum of the bipartite variant $QO$ contains eigenvalue $(-1)$ of multiplicity 3. Thus, in the case of nonbipartite graphs, the answer to the question posed depends on the particular graph $QG$. Here, we continue to explore the spectrum of graphs $QG$. Some possible connections of the mathematical theme to chemistry are also noted. Full article

Spectral graph theory is like a special tool for understanding graphs. It helps us find patterns and connections in complex networks, using the magic of eigenvalues. Let $\mathsf{G}$ be the graph and $\mathsf{A}(\mathsf{G})$ be its adjacency matrix, then $\mathsf{G}$ is singular if the determinant of the adjacency matrix $\mathsf{A}(\mathsf{G})$ is 0, otherwise it is nonsingular. Within the realm of nonsingular graphs, there is the concept of property $\mathtt{R}$, where each eigenvalue’s reciprocal is also an eigenvalue of $\mathsf{G}$. By introducing multiplicity constraints on both eigenvalues and their reciprocals, it becomes property $\mathtt{SR}$. Similarly, the world of nonsingular graphs reveals property $-\mathtt{R}$, where the negative reciprocal of each eigenvalue also finds a place within the spectrum of $\mathsf{G}$. Moreover, when the multiplicity of each eigenvalue and its negative reciprocal is equal, this results in a graph with a property of $-\mathtt{SR}$. Some classes of unweighted nonbipartite graphs are already constructed in the literature with the help of the complete graph ${\mathtt{K}}_n$ and a copy of the path graph ${\mathtt{P}}_4$ satisfying property $\mathtt{R}$ but not $\mathtt{SR}$. This article takes this a step further. The main aim is to construct several weighted classes of graphs which satisfy property $\mathtt{R}$ but not $\mathtt{SR}$. For this purpose, the weight functions are determined that enable these nonbipartite graph classes to satisfy the $-\mathtt{SR}$ and $\mathtt{R}$ properties, even if the unweighted graph does not satisfy these properties. Some examples are presented to support the investigated results. These examples explain how certain weight functions make these special types of graphs meet the properties $\mathtt{R}$ or $-\mathtt{SR}$, even when the original graphs without weights do not meet these properties. Full article

In recent years, the rapid development of the internet and the advancement of information technology have produced a large amount of large-scale data, some of which are presented in the form of large-scale graphs, such as social networks and sensor networks. Minimum vertex cover (MVC) is an important problem in large-scale graph research. This paper proposes a solver Q-MeaMetaVC based on the MVC framework PEAF and the membrane evolution algorithm framework MEAF. First, the graph is reduced and divided into two types of connected components (bipartite graph and non-bipartite graph) to reduce the scale of the problem. Second, different membrane structures are designed for different types of connected components to better represent the connected component features and facilitate solutions. Third, a membrane evolution algorithm (MEA), which includes fusion, division, cytolysis, and selection operators, is designed to solve the connected components. Then, Q-MeaMetaVC is compared with the best MVC solver in recent years on the test set, and good experimental results that are obtained verify the feasibility and effectiveness of Q-MeaMetaVC in solving the MVC of large-scale graphs. Full article

Consensus-based data aggregation in d-regular bipartite graphs poses a challenging task for the scientific community since some of these algorithms diverge in this critical graph topology. Nevertheless, one can see a lack of scientific studies dealing with this topic in the literature. Motivated by our recent research concerned with this issue, we provide a comparative study of frequently applied consensus algorithms for distributed averaging in d-regular bipartite graphs in this paper. More specifically, we examine the performance of these algorithms with bounded execution in this topology in order to identify which algorithm can achieve the consensus despite no reconfiguration and find the best-performing algorithm in these graphs. In the experimental part, we apply the number of iterations required for consensus to evaluate the performance of the algorithms in randomly generated regular bipartite graphs with various connectivities and for three configurations of the applied stopping criterion, allowing us to identify the optimal distributed consensus algorithm for this graph topology. Moreover, the obtained experimental results presented in this paper are compared to other scientific manuscripts where the analyzed algorithms are examined in non-regular non-bipartite topologies. Full article

Let $K[G]$ denote the edge ring of a finite connected simple graph G on $[d]$ and $\mathrm{mat}(G)$ the matching number of G. It is shown that $\mathrm{reg}(K[G])\le \mathrm{mat}(G)$ if G is non-bipartite and $K[G]$ is normal, and that $\mathrm{reg}(K[G])\le \mathrm{mat}(G)-1$ if G is bipartite. Full article