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In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks. [ABSTRACT FROM AUTHOR]

We extend the notion of balance from the realm of signed and gain graphs to conjugate skew gain graphs which are skew gain graphs where the labels on the oriented edges get conjugated when we reverse the orientation. We characterize the balance in a conjugate skew gain graph in several ways especially by dealing with its adjacency matrix and the g -Laplacian matrix. We also deal with the concept of anti-balance in conjugate skew gain graphs. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

The line graph L(G), of a graph G has the edges of G as its vertices and two distinct edges of G are adjacent in L(G), if they are adjacent in G. The Gallai graph (G) of a graph G, has the edges of G as its vertices and two distinct vertices are adjacent in (G) if they are adjacent edges in G, but do not lie on a triangle. The anti-Gallai graph κ(G) of a graph G has the edges of G as its vertices and two distinct edges of G are adjacent in κ(G), if they lie on a common triangle in G. In this paper we find bounds for the eigenvalues of Gallai graph of some class of graphs by using the adjacency spectrum of line graph and anti-Gallai graph. [ABSTRACT FROM AUTHOR]

Key distribution is considered to be among the most significant stages in any cryptographic system, and quantum key distribution is a secure method used to exchange keys between entities involved in communication. In this study, we propose to improve the BB84-quantum key distribution protocol using graph-based coding, where agreement between the two parties is agreed on the graph used in quantum-bases encoding at both ends. This paper presents a simulation of the proposed protocol using python language. The experimental results, which have been very promising, show that the proposed protocol is more effective and secure than the standard BB84 protocol. [ABSTRACT FROM AUTHOR]

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]

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 Aα(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]

This paper proposes a new scheme for very fast detection of a sensor going offline or loss of connectivity of sensor in Wireless Mesh Networks (WMN) in order to avoid any packet loss. A sensor is declared offline by a gateway if it does not receive any packet from the sensor for a specific time interval termed as keep-alive time. In mesh networks, this keep-alive window is kept for a very long duration; sometimes it is as high as 2.5 to 3 hours. The time duration is kept high so that gateway does not declare a sensor that is in deep sleep to conserve battery, as offline. However, keeping a large keep-alive time window leads to the very likely possibility of missing the timely detection of a critical event like a smoke sensor or gas leak sensor going offline. In the proposed method, the gateway maintains its table of neighbourhood matrix in order to facilitate fast and efficient tracking as well as tracing of the nodes for the detection of the loss of connectivity of a sensor. A Laplacian matrix formulation proposed by Fiedler is invoked for tracking and tracing lost nodes of WMN. Through the Fiedler value, this paper substantiates the efficacy of monitoring a large WMN not only for its algebraic connectivity but also to compute the number of disconnected partitions in it. This is achieved through the number of null values of eigenvalues of the Laplacian matrix of the graph of WMN. Through simulation studies, this paper demonstrates achieving an 88 % success rate in the detection of disconnection with an average latency of 39.806 ms. On the contrary, a conventional process for the time duration after which a sensor can be declared offline might be an hour or even more depending upon the pre-set duration of the keep-alive time window. [ABSTRACT FROM AUTHOR]

Nowadays, the process of message encryption is the most important thing to secure our messages and communication between people. Message encryption methods are rapidly increasing currently due to the growth and evolution of the internet and network communications. Sharing information, personal messages, images, or data from one person to another over unsecured channels opens the door for attack, or hacking. To reduce this terminology and to provide better security, cryptographic or encryption techniques play an essential role. We have many kinds of symmetric enciphering techniques like the Caesar Cipher, Atbash Cipher, Hill Cipher, etc., In this paper, we are going to give the enciphering technique with the help of a complete graph, an adjacency matrix, and a generated self-invertible key matrix to encrypt and decrypt the given messages to produce a complicated ciphertext. Since we are using the self-invertible matrix as a key matrix, the inverse of this key matrix is always existing, and while we are decrypting the ciphertext, we do not need to compute the inverse of the key matrix. It helps us to reduce the computational complexity involved in the process of finding the inverse of a key matrix. [ABSTRACT FROM AUTHOR]

In this paper a systematic parametric approach for assessing the water quality for given water resource by using the concepts of graph theory and matrix algebra is presented. A new, novel, generic and more sensitive water quality index (WQI) named as "Baluch" index is proposed that can assess and ranks the water quality for a given water resource. The water quality parameters listed in FAO-29 guidelines (belong to resource under investigation) and their relative importances are utilized to create a water quality assessment graph (WQG) and adjacency matrix. As a case study, the water quality of eighteen stations of upper basin of Indus River, Pakistan is assessed. Eight physicochemical quality parameters including pH, TDS, Total hardness, Ca Hardness, Mg hardness, SO4-2, Cl-1, and EC are used for constructing WQA. This WQA is then used to figure out "Baluch" index for each monitoring station, and it is found that "Baluch" index for Darband station has lowest value i.e. 0.85 and "Baluch" index highlight the situation that all other stations have relatively higher values of pH, which is concern able problem. Furthermore, this new index indicate the lowest quality water belongs to Jijal stations due relatively higher values of pH (i.e. 8.25) and Cl-ion, above average values of TDS, Hardness, SO42-ion concentrations. [ABSTRACT FROM AUTHOR]

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]

A quaternion unit gain graph is a graph where each orientation of an edge is given a quaternion unit, which is the inverse of the quaternion unit assigned to the opposite orientation. In this paper we define the adjacency, Laplacian and incidence matrices for a quaternion unit gain graph and study their properties. These properties generalize several fundamental results from spectral graph theory of ordinary graphs, signed graphs and complex unit gain graphs. Bounds for both the left and right eigenvalues of the adjacency and Laplacian matrix are developed, and the right eigenvalues for the cycle and path graphs are explicitly calculated. [ABSTRACT FROM AUTHOR]

Biomolecular networks are often assumed to be scale-free hierarchical networks. The weighted gene co-expression network analysis (WGCNA) treats gene co-expression networks as undirected scale-free hierarchical weighted networks. The WGCNA R software package uses an Adjacency Matrix to store a network, next calculates the topological overlap matrix (TOM), and then identifies the modules (sub-networks), where each module is assumed to be associated with a certain biological function. The most time-consuming step of WGCNA is to calculate TOM from the Adjacency Matrix in a single thread. In this paper, the single-threaded algorithm of the TOM has been changed into a multi-threaded algorithm (the parameters are the default values of WGCNA). In the multi-threaded algorithm, Rcpp was used to make R call a C++ function, and then C++ used OpenMP to start multiple threads to calculate TOM from the Adjacency Matrix. On shared-memory MultiProcessor systems, the calculation time decreases as the number of CPU cores increases. The algorithm of this paper can promote the application of WGCNA on large data sets, and help other research fields to identify sub-networks in undirected scale-free hierarchical weighted networks. The source codes and usage are available at https://github.com/do-somethings-haha/multi-threaded_calculate_unsigned_TOM_from_unsigned_or_signed_Adjacency_Matrix_of_WGCNA. [ABSTRACT FROM AUTHOR]

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]

Molecular network plays a critical role in determining dynamic elasticity of soft condensed polymers, e.g., their varied cross-linking densities and end-to-end distances with changed Young's moduli. Although it has been studied for decades, the coupling relationship between vertices and edges of molecular networks is not well understood, mainly because the degree of the crosslinking points and asymmetry molecular networks have not been considered in the previously models. In this study, a graph theory was employed to formulate an undirected graphical model and describe the coupling between vertices and edges in molecular networks, based on which the dynamic elasticity of polyelectrolyte hydrogel was modeled. From the Kirchhoff graph theory and bead-spring model, the coupling relationship between vertices and edges was obtained using end-to-end distance and viscosity parameters. Combining the Watts–Strogatz model and kinetic probability, the coupling between vertices and edges for the polyelectrolyte network was studied. Furthermore, an adjacency matrix with eigenvalue, number of vertices and mean degree was proposed to formulate constitutive relationships including dynamic elasticity and stress-strain, according to rubber elasticity theory and Mooney-Rivlin model, respectively. The linking between the vertices and edges determines the network structure and dynamic elasticity of the polyelectrolyte hydrogel. Based on the graph theory, the vertices and edges are encoded by adjacency matrix, which is proposed to describe the dynamic elasticity of symmetric and asymmetric network structures using the crosslinking density and end-to-end distance. Finally, effectiveness of the undirected graphical model was verified using both finite element analysis and experimental results of polyelectrolyte hydrogels reported in literature. Constitutive stress-elongation ratio curves for the polyelectrolyte hydrogel, of which the graph network is determined by the mean degree of d = 2, d = 3, d = 4, d = 5, d = 6 and d = 7, at the same vertex number of N = 8 in molecular network. [Display omitted] • An undirected graphical model and describe the coupling between vertices and edges in molecular networks. • Dynamic elasticity of polyelectrolyte hydrogel has been modeled using the adjacency matrix. • A constitutive relationship is developed to understand the molecular networks and their dynamic elasticities. [ABSTRACT FROM AUTHOR]