Irregular Graphs Research Articles

On the Difference of Mostar Index and Irregularity of Graphs

For a connected graph G, the Mostar index Mo(G) and the irregularity irr(G) are defined as $$Mo(G)=\sum _{uv\in E(G)}|n_u-n_v|$$ and $$ irr (G)=\sum _{uv\in E(G)}|d_u-d_v|$$ , respectively, where $$d_u$$ is the degree of the vertex u of G and $$n_u$$ denotes the number of vertices of G which are closer to u than to v for an edge uv. In this paper, we focus on the difference $$\Delta M(G)=Mo(G)- irr (G)$$ of graphs G. For trees T of order n, we characterize the minimum and second minimum $$\Delta M(T)$$ of T and the minimum $$\Delta M(Tr(T))$$ of the triangulation graphs Tr(T). The parity of $$\Delta M$$ of cactus graphs is also reported. The effect on $$\Delta M$$ is studied for two local operations of subdivision and contraction of an edge in a tree. A formula for $$\Delta M(S(T))$$ of the subdivision trees S(T) and the upper and lower bounds on $$\Delta M(S(T))- \Delta M(T)$$ are determined with the corresponding extremal trees T. Moreover, three related open problems are proposed to $$\Delta M$$ of graphs.

An Irregular Graph Based Network Code for Low-Latency Content Distribution.

To fulfill the increasing demand on low-latency content distribution, this paper considers content distribution using generation-based network coding with the belief propagation decoder. We propose a framework to design generation-based network codes via characterizing them as building an irregular graph, and design the code by evaluating the graph. The and-or tree evaluation technique is extended to analyze the decoding performance. By allowing for non-constant generation sizes, we formulate optimization problems based on the analysis to design degree distributions from which generation sizes are drawn. Extensive simulation results show that the design may achieve both low decoding cost and transmission overhead as compared to existing schemes using constant generation sizes, and satisfactory decoding speed can be achieved. The scheme would be of interest to scenarios where (1) the network topology is not known, dynamically changing, and/or has cycles due to cooperation between end users, and (2) computational/memory costs of nodes are of concern but network transmission rate is spare.

The measure of irregularities of nanosheets

AbstractNanosheets are two-dimensional polymeric materials, which are among the most active areas of investigation of chemistry and physics. Many diverse physicochemical properties of compounds are closely related to their underlying molecular topological descriptors. Thus, topological indices are fascinating beginning points to any statistical approach for attaining quantitative structure–activity (QSAR) and quantitative structure–property (QSPR) relationship studies. Irregularity measures are generally used for quantitative characterization of the topological structure of non-regular graphs. In various applications and problems in material engineering and chemistry, it is valuable to be well-informed of the irregularity of a molecular structure. Furthermore, the estimation of the irregularity of graphs is helpful for not only QSAR/QSPR studies but also different physical and chemical properties, including boiling and melting points, enthalpy of vaporization, entropy, toxicity, and resistance. In this article, we compute the irregularity measures of graphene nanosheet, H-naphtalenic nanosheet, {\text{SiO}}_{2} nanosheet, and the nanosheet covered by {C}_{3} and {C}_{6}.

A new method to predict anomaly in brain network based on graph deep learning.

Functional magnetic resonance imaging a neuroimaging technique which is used in brain disorders and dysfunction studies, has been improved in recent years by mapping the topology of the brain connections, named connectopic mapping. Based on the fact that healthy and unhealthy brain regions and functions differ slightly, studying the complex topology of the functional and structural networks in the human brain is too complicated considering the growth of evaluation measures. One of the applications of irregular graph deep learning is to analyze the human cognitive functions related to the gene expression and related distributed spatial patterns. Since a variety of brain solutions can be dynamically held in the neuronal networks of the brain with different activity patterns and functional connectivity, both node-centric and graph-centric tasks are involved in this application. In this study, we used an individual generative model and high order graph analysis for the region of interest recognition areas of the brain with abnormal connection during performing certain tasks and resting-state or decompose irregular observations. Accordingly, a high order framework of Variational Graph Autoencoder with a Gaussian distributer was proposed in the paper to analyze the functional data in brain imaging studies in which Generative Adversarial Network is employed for optimizing the latent space in the process of learning strong non-rigid graphs among large scale data. Furthermore, the possible modes of correlations were distinguished in abnormal brain connections. Our goal was to find the degree of correlation between the affected regions and their simultaneous occurrence over time. We can take advantage of this to diagnose brain diseases or show the ability of the nervous system to modify brain topology at all angles and brain plasticity according to input stimuli. In this study, we particularly focused on Alzheimer's disease.

On neighborhood distinguishing colorings of graphs

This paper aims at analyzing vertices of proper coloring and several kinds of vertex colorings. The coloring vertices include chief types such as recognizable colorings [1], detectable colorings [2] and, irregular colorings [3]. These colorings are distinctive in some way or other. Regarding irregular colorings, there is a slight deviation which is deemed as Neighborhood distinguishing colorings. This particular coloring has the distinguishing element in that the vertex does not relate to the color assigned to it. The irregular coloring is commonly identified in any sort of graph. This is not the case with the Neighborhood Distinguishing Coloring, since it can be identified in a graph under the condition that if and only if any two non-adjacent vertices do not have the same neighbor. The basic definition and the properties related to this Neighborhood Distinguishing Coloring are illustrated in [5]. Further, this paper attempts to determine the NDC-number of sum of the graphs, Corona of a graph and neighborhood irregular graphs.

Modelling Networks as Neighborly Irregular Graphs

Today’s world is filled with numerous computing devices and electronic gadgets connected to the Internet. These devices continuously sense and deliver their desired task autonomously or via owners. Domestic applications like healthcare monitoring systems, smart farming, noise pollution control, etc., involves many sensors and wearables that tirelessly estimate, evaluate and report the desired outcome. There are circumstances where two or more sensors at the same level, sense the same information and rely to the controller/base station. Handling and processing of such duplicate information results in increased overhead messages and reduced lifetime of sensors. This paper proposes a novel idea to model a network as a Neighborly Irregular graph so that optimal placement of sensor nodes can be guaranteed and communication between sensors at the same level is restricted. A simple and novel algorithm to construct Neighborly Irregular graph is proposed which converts the underlying network to a Neighborly Irregular graph if the network is not Neighborly Irregular. The proposed idea is tested with smart irrigation system in real time to prove its effectiveness. Experimental results prove that the message overheads are drastically reduced when the underlying network is Neighborly Irregular.

Special Classes of Coprime Irregular Graphs

: An k-edge-weighting of a graph G = (V, E) is a mapping : E(G) {1, 2, 3, ...k}, where k is a positive integer. The sum of the edge-weighting appearing on the edges incident at the vertex v under the edge-weighting and is denoted by (v). An k edge-weighting of G is a coprime irregular edge-weighting if gcd ( (u), (v)) = 1 for every pair of adjacent vertices u and v in G. A graph G admits a coprime irregular edge-weighting is called a coprime irregular graph. In this paper, we discuss the coprime irregular edge-weighting for some special classes of graphs.

A Relation between D-Index and Wiener Index for r-Regular Graphs

For any two distinct vertices u and v in a connected graph G, let lPu,v=lP be the length of u−v path P and the D–distance between u and v of G is defined as: dDu,v=minplP+∑∀y∈VPdeg y, where the minimum is taken over all u−v paths P and the sum is taken over all vertices of u−v path P. The D-index of G is defined as WDG=1/2∑∀v,u∈VGdDu,v. In this paper, we found a general formula that links the Wiener index with D-index of a regular graph G. Moreover, we obtained different formulas of many special irregular graphs.

On Equitable Irregular Graphs

An k−edge-weighting of a graph G = (V,E) is a map 𝝋: 𝑬(𝑮) → {𝟏,𝟐,𝟑, . . . 𝒌}, }where 𝒌 ≥ 𝟏 is an integer. Denote 𝑺𝝋(𝒗) is the sum of edge-weights appearing on the edges incident at the vertex v under𝝋 . An k-edge -weighting of G is equitable irregular if |𝑺𝝋(𝒖) − 𝑺𝝋(𝒗)| ≤ 𝟏, for every pair of adjacent vertices u and v in G. The equitable irregular strength 𝑺𝒆 (𝑮) of an equitable irregular graph G is the smallest positive integer k such that there is a k-edge weighting of G. In this paper, we discuss the equitable irregular edge-weighting for some classes of graphs

Characterising spatial dependence on epidemic thresholds in networks

Epidemic processes are an important security research topic for both the internet and social networks. The epidemic threshold is a fundamental metric used to evaluate epidemic spread in networks. Previous work has shown that the epidemic threshold of a network is 1/λmax(A), i.e., the inverse of the largest eigenvalue of its adjacency matrix. In this work, however, we indicate that such a theoretical threshold ignores spatial dependence among nodes and hence underestimates the actual epidemic threshold. Moreover, inspired by the Markov random field, we analytically derive a more accurate epidemic threshold based on a spatial Markov dependence assumption. Our model shows that the epidemic threshold is indeed 1/λmax(A)(1 − ρ), where ρ is the average spatial correlation coefficient between neighbouring nodes. We then apply simulations to compare the performance of these two theoretical epidemic thresholds in different networks, including regular graphs, synthesised irregular graphs, and a real topology. We find that our proposed epidemic threshold incorporates a certain spatial dependence and thus achieves greater accuracy in characterising the actual epidemic threshold in networks.

On irregularity descriptors of derived graphs

Topological indices are molecular structural descriptors which computationally and theoretically describe the natures of the underlying connectivity of nanomaterials and chemical compounds, and hence they provide quicker methods to examine their activities and properties. Irregularity indices are mainly used to characterize the topological structures of irregular graphs. Graph irregularity studies are useful not only for quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) studies, but also for predicting their various physical and chemical properties, including toxicity, resistance, melting and boiling points, the enthalpy of evaporation and entropy. In this article, we establish the expressions for the irregularity indices named as the variance of vertex degrees, σ irregularity index, and the discrepancy index of subdivision graph, vertex-semi total graph, edge-semi total graph, total graph, line graph, paraline graph, double graph, strong double graph and extended double cover of a graph.

On the α-spectral radius of graphs

For 0 ? ? ? 1, Nikiforov proposed to study the spectral properties of the family of matrices A?(G) = ?D(G)+(1 ? ?)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The ?-spectral radius of G is the largest eigenvalue of A?(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the ?-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the ?-spectral radius for unicyclic graphs G with maximum degree ? ? 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest ?-spectral radius among trees, and the unique tree with the largest ?-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the ?-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively.

Stepwise transmission irregular graphs

The distance d(u, v) between vertices u and v of a connected graph G is defined as the number of edges in a shortest path connecting them. The transmission of a vertex v of G is the sum of distances from v to all the other vertices of G. A graph is stepwise transmission irregular (STI) if the transmissions of any two of its adjacent vertices differ by exactly one. Some basic properties of STI graphs are established and infinite families are constructed.

Recent results of quantum ergodicity on graphs and further investigation

We outline some recent proofs of quantum ergodicity on large graphs and give new applications in the context of irregular graphs. We also discuss some remaining questions.

Note on non-regular graphs with minimal total irregularity

Let G be a graph with vertex set V(G). The total irregularity of G is defined as irrt(G)=∑{u,v}⊆V(G)|degG(u)−degG(v)|, where degG(v) is the degree of the vertex v of G. The aim of this paper is to present some bounds for this graph invariant. A new simple proof for a recently proposed conjecture on total irregularity of graphs is also presented.

Cospectral Bipartite Graphs with the Same Degree Sequences but with Different Number of Large Cycles

Finding the multiplicity of cycles in bipartite graphs is a fundamental problem of interest in many fields including the analysis and design of low-density parity-check (LDPC) codes. Recently, Blake and Lin computed the number of shortest cycles (g-cycles, where g is the girth of the graph) in a bi-regular bipartite graph, in terms of the degree sequences and the spectrum (eigenvalues of the adjacency matrix) of the graph (Blake and Lin in IEEE Trans Inf Theory 64(10): 6526–6535, 2018). This result was subsequently extended in Dehghan and Banihashemi (IEEE Trans Inf Theory 65(6):3778–3789, 2019) to cycles of length $$g+2, \ldots , 2g-2$$, in bi-regular bipartite graphs, as well as 4-cycles and 6-cycles in irregular and half-regular bipartite graphs, with $$g \ge 4$$ and $$g \ge 6$$, respectively. In this paper, we complement these positive results with negative results demonstrating that the information of the degree sequences and the spectrum of a bipartite graph is, in general, insufficient to count (a) the i-cycles, $$i \ge 2g$$, in bi-regular graphs, (b) the i-cycles for any $$i > g$$, regardless of the value of g, and g-cycles for $$g \ge 6$$, in irregular graphs, and (c) the i-cycles for any $$i > g$$, regardless of the value of g, and g-cycles for $$g \ge 8$$, in half-regular graphs. To obtain these results, we construct counter-examples using the Godsil–McKay switching.

Tripartite graphs with energy aggregation

The aggregate of the absolute values of the graph eigenvalues is called the energy of a graph. It is used to approximate the total _-electron energy of molecules. Thus, finding a new mechanism to calculate the total energy of some graphs is a challenge; it has received a lot of research attention. We study the eigenvalues of a complete tripartite graph Ti,i,n−2i , for n _ 4, based on the adjacency, Laplacian, and signless Laplacian matrices. In terms of the degree sequence, the extreme eigenvalues of the irregular graphs energy are found to characterize the component with the maximum energy. The chemical HMO approach is particularly successful in the case of the total _-electron energy. We showed that some chemical components are equienergetic with the tripartite graph. This discovering helps easily to derive the HMO for most of these components despite their different structures.

Fuzzy Graphs

In this paper, neighbourly irregular fuzzy graphs, neighbourly total irregular fuzzy graphs, highly irregular fuzzy graphs and highly total irregular fuzzy graphs are introduced. A necessary and sufficient condition under which neighbourly irregular and highly irregular fuzzy graphs are equivalent is provided. We define d2 degree of a vertex in fuzzy graphs and total d2 -degree of a vertex in fuzzy graphs and (2, k)-regular fuzzy graphs, totally (2, k)- regular fuzzy graphs are introduced. (2, k)- regular fuzzy graphs and totally (2, k)-regular fuzzy graphs are compared through various examples.

Irregularity of Sierpinski graph

A graph is regular if all of its vertices have equal degree and otherwise, it is an irregular graph. Irregularity topological indices are used for studying irregularly of irregular graphs. These indices are used in different problems of physics, engineering and chemistry. Moreover, irregularity of molecules is useful in for QSAR/QSPR studies. In this paper, we calculated different irregularity indices for Sierpinski Graph.

Improving Efficiency of Parallel Vertex-Centric Algorithms for Irregular Graphs

Memory access is known to be the main bottleneck for shared-memory parallel graph applications especially for large and irregular graphs. Propagation blocking (PB) idea was proposed recently to improve the parallel performance of PageRank and sparse matrix and vector multiplication operations. The idea is based on separating parallel computation into two phases, binning and accumulation, such that random memory accesses are replaced with contiguous accesses. In this paper, we propose an algorithm that allows execution of these two phases concurrently. We propose several improvements to increase parallel throughput, reduce memory overhead, and improve work efficiency. Our experimental results show that our proposed algorithms improve shared-memory parallel throughput by a factor of up to 2x compared to the original PB algorithms. We also show that the memory overhead can be reduced significantly (from 170 percent down to less than 5 percent) without significant degradation of performance. Finally, we demonstrate that our concurrent execution model allows asynchronous parallel execution, leading to significant work efficiency in addition to throughput improvements.