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The common divisor graph, Γ (X) , is a graph that has been defined on a set of positive integers X. Some properties of this graph have been studied in the cases when either X is the set of character degrees or the set of conjugacy class sizes of a group. This paper deals with several properties of a particular case of the common divisor graph, Γ (Z) , when Z is a multiset of positive integers that admits a decomposition Z = X Y , where X Y = { x y | x ∈ X , y ∈ Y } and 1 ∈ X and 1 ∈ Y . Our results can be applied for the graphs associated to character degrees and conjugacy classes of the direct product of two finite groups. [ABSTRACT FROM AUTHOR]

This paper presents tight upper bounds for all of the signless Laplacian eigenvalues of a graph with prescribed order and minimum degree, which improve previously known upper bounds. Also, the relation between the number of signless Laplacian eigenvalues falling within specific intervals and various graph parameters such as independence, clique, chromatic, edge covering and matching numbers are explored. [ABSTRACT FROM AUTHOR]

Multisource switched-capacitor multilevel inverters are proper topologies when multiple dc sources are available as renewable energy sources. However, these multilevel inverters have the main drawbacks of requiring a large number of power semiconductors and capacitors and a low boost factor to achieve a large number of voltage levels, which complicate control, reduce performance, and raise the cost. This article presents two configurations for multisource switched-capacitor multilevel inverters. First, a basic multisource switched-capacitor topology is proposed with a small number of capacitors and semiconductors that produce a zero voltage level and a large number of positive voltage levels. The input dc sources are operated in an asymmetric trinary algorithm, while the capacitors are charged/discharged in a binary algorithm using parallel/series modes (without the need for any circuit balancing). Based on the proposed basic topology, two multilevel inverter topologies are proposed, which create bipolar output voltage levels using two different circuits at the output. The first proposed topology uses a standard H-bridge inverter at the output of the basic topology, while the second topology uses an extra switched-capacitor (self-balancing) circuit. The comparison indicates that both the proposed topologies have advantages over most of the reported multisource switched-capacitor multilevel inverters, such as a small number of capacitors, a small number of power semiconductors, low voltage stress, and high boost factor. Furthermore, the proposed topologies decrease the low cost factor in comparison to other topologies. The capacitors' capacitance, power losses, and voltage ripple losses are all measured and evaluated in detail. Theoretical research is validated with the use of laboratory prototypes to validate the performance of the proposed topologies. [ABSTRACT FROM AUTHOR]

Given a finite group G , the character graph , denoted by Δ (G) , for its irreducible character degrees is a graph with vertex set ρ (G) which is the set of prime numbers that divide the irreducible character degrees of G , and with { p , q } being an edge if there exists a non-linear χ ∈ Irr (G) whose degree is divisible by p q. In this paper, on one hand, we proceed by discussing the graphical shape of Δ (G) when it has cut vertices or small number of eigenvalues, and on the other hand we give some results on the group structure of G with such Δ (G). Recently, Lewis and Meng proved the character graph of each solvable group has at most one cut vertex. Now, we determine the structure of character graphs of solvable groups with a cut vertex and diameter 3. Furthermore, we study solvable groups whose character graphs have at most two distinct eigenvalues. Moreover, we investigate the solvable groups whose character graphs are regular with three distinct eigenvalues. In addition, we give some lower bounds for the number of edges of Δ (G). [ABSTRACT FROM AUTHOR]

Abstract In a network or a graph, the total communicability (TC) has been defined as the sum of the entries in the exponential of the adjacency matrix of the network. This quantity offers a good measure of how easily information spreads across the network, and can be useful in the design of networks having certain desirable properties. In this paper, we obtain some bounds for total communicability of a graph G , T C (G) , in terms of spectral radius of the adjacency matrix, number of vertices, number of edges, minimum degree and the maximum degree of G. Moreover, we find some upper bounds for T C (G) when G is the Cartesian product, tensor product or the strong product of two graphs. In addition, Nordhaus–Gaddum-type results for the total communicability of a graph G are established. [ABSTRACT FROM AUTHOR]

The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of degrees of vertices of G. In this paper we investigate its behavior under several binary operations on graphs. We present explicit formulas for its values for composite graphs in terms of its values for operands and some auxiliary invariants. We demonstrate applications of our results to several chemically relevant classes of graphs and show how the Narumi-Katayama index can be used as a measure of graph irregularity. [ABSTRACT FROM AUTHOR]

In this work, the (2+1)-dimensional chiral nonlinear Schrödinger equation that describes about quantum field concept in physics and other physical sciences are studied and solved by utilizing the two modern techniques including the polynomial expansion method and the Sardar sub-equation method. We attained different types of soliton solutions that had been applications in different fields of mathematical sciences. The behaviours of attained solutions are periodic, singular and v-shaped soliton solutions. Furthermore, we have investigated the stability of the obtained results. Also, the 3D, 2D, and contour graphics are displayed for the better understanding of the dynamical behaviour of various waves structures extensively. The techniques applied in this article are not used in this model in literature so we say that our findings are new that summarize the novelty of work. The utilize model has applications in physics related phenomenon also obtained results highly valuable in various branches of sciences specially in the transmission of fiber optical. [ABSTRACT FROM AUTHOR]

Nonlinear phenomena, characterized by behaviors that cannot be explained by linear systems and has significant challenges in understanding and modeling. To address this, the mathematical description of such phenomena relies on differential equations. In this study, we investigate the cubic-quintic-septimal nonlinear (7th order nonlinear media) Schrödinger wave equation, which governs the evolution of light beams in a weak non-local medium. The novelty of our study lies in the application of the improved generalized Riccati equation mapping method to obtain exact solutions for the governed equation. This scheme offers a systematic and reliable approach to exploring nonlinear phenomena, contributing to the advancement of nonlinear science and its practical applications. By applying the proposed scheme, a range of exact solutions encompassing trigonometric, rational, exponential, and hyperbolic functions are derived which offer insights into the dynamics of the light beams. Additionally, 2D and 3D graphical illustration are presented to provide a comprehensive demonstration of their dynamical behavior. Furthermore, it is important to highlight the significance of studying the cubic-quintic-septimal nonlinear Schrödinger wave equation which has application in various domains such as quantum mechanics, optics, and nonlinear wave propagation. Understanding of its solutions facilitates the design and optimization of systems involving weak non-local media. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to study various optical soliton profiles of the Schrödinger–Hirota (SH) equation having the parabolic law nonlinearity by utilizing the Sardar-Subequation (SSe) method. First, the general method of the suggested SSe method is presented. Then using traveling wave transformation the suggested equation is converted into the nonlinear ordinary differential equation, where the real and imaginary parts are separated. Furthermore, the proposed method is utilized and novel optical solitons of the suggested SH equation with parabolic law are obtained. Furthermore, the solutions are validated through stability analysis. The exact analytical solutions are graphically simulated where the effects of different parameters on the soliton wave profiles are studied. [ABSTRACT FROM AUTHOR]

In this article, the extended fan method and the positive quadratic function technique are used to solve the (2 + 1) -dimensional Chiral nonlinear Schrödinger equation. Several new outcomes are obtained. Several new results are obtained. We can offer soliton-type solutions, triangular-type solutions, simple and combined solutions using Jacobi's elliptic functions. Using suitable parameters, some graphics are given to observe the evolution of these analytical solutions. Many other nonlinear differential equations can benefit from using the methods proposed in this work. [ABSTRACT FROM AUTHOR]

With the developing technology, magneto-optical and ferromagnetic materials are gaining importance and are used especially in magneto-optics, ferromagnetism, fluid mechanics, etc. These processes are modeled via Kadomtsev–Petviashvili-type models. In this work, a generalized (3+1)-dimensional variable-coefficient modified Kadomtsev–Petviashvili (vcmKP) system and special cases are considered that simulates electromagnetic, water, and powder-acoustic/ion-acoustic/dust-ion-acoustic waves. As to the novelty of this paper, the travelling wave, soliton solutions of the considered systems are hold by using Bernoulli method which is the well-known ansatz-based method and the analytical method. As far as we know, the obtained solutions are seen for the first time in this study and are important for the development of the use of magneto-optical and ferromagnetic materials in industry and applied sciences, fiber optic communication fields. [ABSTRACT FROM AUTHOR]

This study proposed a modified G′/G2-expansion method to seek the new exact traveling wave solutions of the Newell–White–Head–Segel (NWHS) Model. This is an amplitude equation utilized for distributing temperature within a rod that is infinitely thin and long, or determining the flow velocity of a fluid through a pipe that is infinitely long but has a small diameter. The modified G′/G2-expansion method is used to extract the new exact solutions. The solutions of this model are categorized in hyperbolic, trigonometric, and rational forms. Moreover, we compare our results with the new auxiliary equation method. The 3D, line and corresponding contour representation of these solutions are depicted by choosing the different values of parameters. [ABSTRACT FROM AUTHOR]

The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is investigated in this study. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is an extension of Calogero–Bogoyavlenskii–Schiff equation that describes the movement of Riemann waves along y-axis while long waves moves along the x-axis. The dynamics of Riemann waves is one of the most significant applications including tsunami in rivers, internal waves in oceans and magento-sound waves in plasmas. Finding new precise solutions with the assistance of a relatively new extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique is the primary objective of this effort. The suggested techniques are important tools in the fields of mathematical physics. Successful extraction of hyperbolic, rational, and trigonometric function solutions are achieved by using the proposed analytical methods. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is studied for the first time using extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique in this work and novel solutions are observed. 3D plots, contour plots and 2D plots are used to depict the dynamics of the extracted solutions. [ABSTRACT FROM AUTHOR]

This paper is mainly concerning the study of truncated M-fractional Kuralay equations that have applications in numerous fields, including nonlinear optics, ferromagnetic materials, signal processing, engineering fields and optical fibers. Due to its ability to clarify a wide range of sophisticated physical phenomena and reveal more dynamic structures of localized wave solutions, the Kuralay equation has captured a lot of attention in the research field. The newly designed integration methods, known as the modified Sardar subequation method and enhanced modified extended tanh expansion method are used as solving tools to validate the solutions. The goal of this study is to extract several kinds of optical solitons, such as mixed, dark, singular, bright-dark, bright, complex and combined solitons. Due to the many potential applications for superfast signal routing techniques and shorter light pulses in communications, the optical propagation of soliton in optical fibers is now a topic of significant interest. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. In addition, exponential, periodic, hyperbolic solutions are generated. The applied approaches are efficient in explaining fractional nonlinear partial differential equations by providing pre-existing solutions and also producing new solutions by combining results from multiple processes. Additionally, we plot the contour, 2D, and 3D graphs with the associated parameter values to visualize the solutions. The results of this study show the effectiveness of the approaches adopted and help enhance comprehension of the nonlinear dynamical behavior of specific systems. We expect that a substantial amount of engineering model specialists will greatly benefit from our work. The findings demonstrate the efficacy, efficiency, and applicability of the computational method employed, particularly in dealing with intricate systems. [ABSTRACT FROM AUTHOR]

The singular values of a matrix A are defined as the square roots of the eigenvalues of A ⁎ A , and the energy of A denoted by E (A) is the sum of its singular values. The energy of a graph G , E (G) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A = ( B D D ⁎ C ) , then E (A) ≥ 2 E (D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E (H) is an edge cut of G , then E (H) ≤ E (G) , i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E (G) ≥ 4 m + n (n − 2) | det ⁡ (A) | 2 n . Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E (G) given in Oboudi (2019) [14]. [ABSTRACT FROM AUTHOR]

The energy of a graph G, denoted by E(G), is defined as the sum of absolute values of all eigenvalues of G. A graph G is called reciprocal if 1/≥ is an eigenvalue of G whenever λ is an eigenvalue of G. Further, if λ and 1 λ have the same multiplicities, for each eigenvalue λ, then it is called strong reciprocal. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631{633), it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree ∞(G) whose adjacency matrix is non-singular, E(G) λ Δ(G) + ∞(G) and the equality holds if and only if G is a complete graph. Here, we prove the validity of this conjecture for some strong reciprocal graphs. Moreover, we show that if G is a strong reciprocal graph, then E(G) λ Δ(G) + ∞(G) - 1 2. Recently, it has been proved that if G is a reciprocal graph of order n and its spectral radius, ?, is at least 4λmin, where λmin is the smallest absolute value of eigenvalues of G, then E(G) λ n + 1 2. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption. [ABSTRACT FROM AUTHOR]

A graph is called equimatchable if all of its maximal matchings have the same size. Kawarabayashi, Plummer, and Saito showed that the only connected equimatchable 3-regular graphs are K4 and K3, 3. We extend this result by showing that for an odd positive integer r, if G is a connected equimatchable r-regular graph, then [ABSTRACT FROM AUTHOR]

Let G be a graph. An edge orientation of G is called smooth if the in-degree and the out-degree of every vertex differ by at most one. In this paper, we show that if G is a 2-edge-connected non-bipartite graph with δ ( G ) ≥ 3 , then G has a smooth primitive orientation. Among other results, using the spectral radius of digraphs, we show that if D 1 is a primitive regular orientation and D 2 is a non-regular orientation of a given graph, then for sufficiently large t , the number of closed walks of length t in D 1 is more than the number of closed walks of length t in D 2 . [ABSTRACT FROM AUTHOR]

The energy of a graph G , E (G) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In Akbari and Hosseinzadeh (2020) [3] it was conjectured that for every graph G with maximum degree Δ (G) and minimum degree δ (G) whose adjacency matrix is non-singular, E (G) ≥ Δ (G) + δ (G) and the equality holds if and only if G is a complete graph. Let G be a connected graph with the edge set E (G). In this paper, first we show that E (L (G)) ≥ | E (G) | + Δ (G) − 5 , where L (G) denotes the line graph of G. Next, using this result, we prove the validity of the conjecture for the line of each connected graph of order at least 7. [ABSTRACT FROM AUTHOR]

A Deza graph G with parameters (n , k , b , a) is a k -regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children G A and G B of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in G A or G B if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular. [ABSTRACT FROM AUTHOR]