12 results on '"Adjacency matrix"'
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2. BOUNDS FOR THE α-ADJACENCY ENERGY OF A GRAPH.
- Author
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SHABAN, REZWAN UL, IMRAN, MUHAMMAD, and GANIE, HILAL A.
- Subjects
GRAPH theory ,EIGENVALUES ,CONVEX functions ,RAYLEIGH quotient ,GRAPH connectivity - Abstract
For the adjacency matrix A(G) and diagonal matrix of the vertex degrees D(G) of a simple graph G, the A(G) matrix is the convex combinations of D(G) and A(G), and is defined as A(G) = D(G)+(1)A(G), for 0 n be the eigenvalues of A(G) (which we call -adjacency eigenvalues of the graph G). The generalized adjacency energy also called -adjacency energy of the graph G is defined as EA (G) = is the average vertex degree, m is the size and n is the order of G. The -adjacency energy of a graph G merges the theory of energy (adjacency energy) and the signless Laplacian energy, as EA0 (G) = E (G) and 2E A 12 (G) = QE(G), where E (G) is the energy and QE(G) is the signless Laplacian energy of G. In this paper, we obtain some new upper and lower bounds for the generalized adjacency energy of a graph, in terms of different graph parameters like the vertex covering number, the Zagreb index, the number of edges, the number of vertices, etc. We characterize the extremal graphs attained these bounds. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. Some Properties of Double Domination in Vague Graphs with an Application.
- Author
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Rao, Yongsheng, Cai, Ruiqi, Talebi, Ali Asghar, and Mojahedfar, Masomeh
- Subjects
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GRAPH theory , *DOMINATING set , *FUZZY graphs , *ENERGY consumption , *SYMMETRY - Abstract
This paper is devoted to the study of the double domination in vague graphs, and it is a contribution to the Special Issue "Advances in graph theory and Symmetry/Asymmetry" of Symmetry. Symmetry is one of the most important criteria that illustrate the structure and properties of fuzzy graphs. It has many applications in dominating sets and helps find a suitable place for construction. Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In the graph theory, a dominating set (DS) for a graph G * = (X , E) is a subset D of the vertices X so that every vertex which is not in D is adjacent to at least one member of D. The subject of energy in graph theory is one of the most attractive topics serving a very important role in biological and chemical sciences. Hence, in this work, we express the notion of energy on a dominating vague graph (DVG) and also use the concept of energy in modeling problems related to DVGs. Moreover, we introduce a new notion of a double dominating vague graph (DDVG) and provide some examples to explain various concepts introduced. Finally, we present an application of energy on DVGs. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
4. Some New Bounds for α -Adjacency Energy of Graphs.
- Author
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Zhang, Haixia and Zhang, Zhuolin
- Subjects
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GRAPH theory , *EIGENVALUES , *STRUCTURAL analysis (Engineering) , *ENERGY consumption - Abstract
Let G be a graph with the adjacency matrix A (G) , and let D (G) be the diagonal matrix of the degrees of G. Nikiforov first defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G) , 0 ≤ α ≤ 1 , which shed new light on A (G) and Q (G) = D (G) + A (G) , and yielded some surprises. The α − adjacency energy E A α (G) of G is a new invariant that is calculated from the eigenvalues of A α (G) . In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for E A α (G) in terms of graph parameters (the order n, the edge size m, etc.) and characterize the corresponding extremal graphs. In addition, we obtain some relations between E A α (G) and other energies such as the energy E (G) . Some results can be applied to appropriately estimate the α -adjacency energy using some given graph parameters rather than by performing some tedious calculations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
5. Spectrum of prism graph and relation with network related quantities.
- Author
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Raza, Ali, Munir, Mobeen, Abbas, Tasawar, Eldin, Sayed M., and Khan, Ilyas
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GRAPH theory ,STABILITY theory ,POLYHEDRA ,SPANNING trees ,RADIUS (Geometry) - Abstract
Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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6. Graph Theory: A Lost Component for Development in Nigeria.
- Author
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Babarinsa, Olayiwola
- Subjects
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GRAPH theory , *LAPLACIAN matrices , *MATHEMATICS , *RESEARCH - Abstract
Graph theory is one of the neglected branches of mathematics in Nigeria but with the most applications in other fields of research. This article shows the paucity, importance, and necessity of graph theory in the development of Nigeria. The adjacency matrix and dual graph of the Nigeria map were presented. The graph spectrum and energies (graph energy and Laplacian energy) of the dual graph were computed. Then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the Eulerian circuit and Hamiltonian paths from the dual graph were obtained and discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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7. Aα matrix of commuting graphs of non-abelian groups.
- Author
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Rather, Bilal A., Ali, Fawad, Ullah, Nasim, Mohammad, Al-Sharef, Din, Anwarud, and Sehra
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EIGENVALUES ,EIGENANALYSIS ,EXPONENTIAL stability ,LAPLACIAN matrices ,GRAPH theory - Abstract
For a finite group G and a subset X≠ 0 of G, the commuting graph, indicated by G = C(G, X), is the simple connected graph with vertex set X and two distinct vertices x and y are edge connected in G if and only if they commute in X. The Aα matrix of G is specified as A
α (G) = αD(G) + (1-α)A(G), α ∈ [0, 1], where D(G) is the diagonal matrix of vertex degrees while A(G) is the adjacency matrix of G. In this article, we investigate the Aα matrix for commuting graphs of finite groups and we also find the Aα eigenvalues of the dihedral, the semidihedral and the dicyclic groups. We determine the upper bounds for the largest Aα eigenvalue for these graphs. Consequently, we get the adjacency eigenvalues, the Laplacian eigenvalues, and the signless Laplacian eigenvalues of these graphs for particular values of α. Further, we show that these graphs are Laplacian integral. [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
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8. Two upper bounds on the Aα-spectral radius of a connected graph.
- Author
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Pirzada, Shariefuddin
- Subjects
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GRAPH theory , *GEOMETRIC vertices , *EIGENVALUES , *SUBGRAPHS , *GENERALIZATION - Abstract
If A(G) and D(G) are respectively the adjacency matrix and the diagonal matrix of vertex degrees of a connected graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G) = α D(G) + (1 - α) A(G), where 0 ≤ α ≤ 1. The Aα (or generalized) spectral radius λ(Aα(G)) (or simply λα) is the largest eigenvalue of Aα(G). In this paper, we show that λα ≤ α Δ + (1 - α)√ 2m ( 1 - 1/ω), where m, Δ and ω = ω(G) are respectively the size, the largest degree and the clique number of G. Further, if G has order n, then we show that ... where di and mi are respectively the degree and the average 2-degree of the vertex Vi. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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9. Change of nullity of a graph under two operations.
- Author
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Mohiaddin, Gohdar H. and Sharaf, Khidir R.
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INDEPENDENT variables , *GRAPH theory , *EIGENVALUES , *MULTIPLICITY (Mathematics) - Abstract
The nullity of a graph G is the multiplicity of zero as an eigenvalue of its adjacency matrix. An assignment of weights to the vertices of a graph, that satisfies a zero sum condition over the neighbors of each vertex, and uses maximum number of independent variables is denoted by a high zero sum weighting of the graph. This applicable tool is used to determine the nullity of the graph. Two types of graphs are defined, and the change of their nullities is studied, namely, the graph G+ab constructed from G by adding a new vertex ab which is joint to all neighbors of both vertices a and b of G, and G•ab which is obtained from G+ab by removing both vertices a and b. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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10. A spherical evolution algorithm with two-stage search for global optimization and real-world problems.
- Author
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Wang, Yirui, Cai, Zonghui, Guo, Lijun, Li, Guoqing, Yu, Yang, and Gao, Shangce
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GLOBAL optimization , *SEARCH algorithms , *TIME complexity , *GRAPH theory , *COMPUTATIONAL complexity - Abstract
This paper proposes a spherical evolution algorithm with two-stage search. Spherical search and hypercube search are combined to achieve individuals' evolution. A self-adaptive Gaussian scale factor and a variable scale factor are designed to adaptively control individuals' spherical and hypercube search area according to their search situations. Two search stages frequently switch with four search cases to enhance the balance between exploration and exploitation processes. A directed adjacency matrix is devised to analyze the relationship among individuals from the perspective of graph theory. Experiments compare the proposed algorithm with five algorithms with distinctive search patterns on twenty nine CEC2017 benchmark functions. The diversity analysis and graph theory analysis show the good population diversity and effective information spreading of the proposed algorithm. Twenty two real-world problems evaluate the practicality and optimization ability of the proposed algorithm. Finally, the computational time complexity demonstrates that the proposed algorithm is more efficient than the original spherical evolution algorithm. • A spherical evolution algorithm with two-stage search is proposed. • Spherical search and hypercube search are combined to adaptively guide individuals' evolution. • A directed adjacency matrix is firstly introduced to analyze the relationship among individuals. • Functions and real-world problems verify the superior search performance of the proposed algorithm. • The computational efficiency of the proposed algorithm is enhanced. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
11. A Poisson multi-Bernoulli mixture filter for tracking multiple resolvable group targets.
- Author
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Zhou, Yusong, Zhao, Jin, Wu, Sunyong, and Liu, Chang
- Subjects
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FILTERS & filtration , *GRAPH theory , *MIXTURES , *FUNCTIONALS , *DYNAMIC models - Abstract
This paper presents a novel Poisson multi-Bernoulli mixture (PMBM) filter for tracking multiple resolvable group targets (MRGT) based on graph theory. Firstly, the number of groups and the relationships between members within the group are modelled by the adjacency matrix. Then, considering that a single dynamic evolution model is insufficient to guarantee stable tracking performance for group targets, the virtual leader-follower model (VLFM) is introduced to predict the evolution trend of unknown and potentially detected targets, respectively. Furthermore, we prove the conjugation of the proposed algorithm with the probability generating functionals (PGF) and give a detailed implementation of the Gaussian mixture (GM). Based on the coexistence scenario of splitting, merging and non-linear motion of the group targets, the simulation results show the effectiveness of the proposed algorithm in comparison with the existing methods. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
12. Cospectral graphs : What properties are determined by the spectrum of a graph?
- Author
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Sundström, Erik
- Subjects
graphs ,graph spectrum ,Matematik ,cospectral graphs ,adjacency matrix ,spectra ,graph theory ,adjacency matrices ,graph spectra ,graph ,spectral graph theory ,Mathematics ,spectrum - Abstract
This paper was written as a bachelor thesis in mathematics. We study adjacency matrices and their eigenvalues to investigate what properties of the corresponding graphs can be determined by those eigenvalues, the spectrum of the graph. The question of which graphs are uniquely determined by their spectra is also covered. Later on we study some methods of finding examples of graphs with shared spectra, also referred to as cospectral graphs.
- Published
- 2023
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