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In the modern era of the chemical science, the chemical graph theory has contributed significantly to exploring the properties of the chemical compounds. Currently, the computation of the topological indices is one of the most active directions of the research in the area of the chemical graph theory. The main feature of the study of the topological indices is its its ability of predicting the various physio-chemical properties. In this article, we compute several degree-based topological indices for the caboxy-terminated dendritic macromolecule. We compute Harmonic index, atom-bond connectivity index, geometric arithmetic index, sum connectivity index, inverse sum index, symmetric division degree, and Zagreb indices for caboxy-terminated dendritic macromolecule. The obtained results have potential to predict biochemical properties such as viscosity, entropy, and boiling point.

Cubic fuzzy graphs (CFGs) offer greater utility as compared to interval-valued fuzzy graphs and fuzzy graphs due to their ability to represent the degree of membership for vertices and edges using both interval and fuzzy number forms. The significance of these concepts motivates us to analyze and interpret intricate networks, enabling more effective decision making and optimization in various domains, including transportation, social networks, trade networks, and communication systems. This paper introduces the concepts of vertex and edge connectivity in CFGs, along with discussions on partial cubic fuzzy cut nodes and partial cubic fuzzy edge cuts, and presents several related results with the help of some examples to enhance understanding. In addition, this paper introduces the idea of partial cubic α-strong and partial cubic δ-weak edges. An example is discussed to explain the motivation behind partial cubic α-strong edges. Moreover, it delves into the introduction of generalized vertex and edge connectivity in CFGs, along with generalized partial cubic fuzzy cut nodes and generalized partial cubic fuzzy edge cuts. Relevant results pertaining to these concepts are also discussed. As an application, the concept of generalized partial cubic fuzzy edge cuts is applied to identify regions that are most affected by trade deficits resulting from street crimes. Finally, the research findings are compared with the existing method to demonstrate their suitability and creativity. [ABSTRACT FROM AUTHOR]

In this paper, a new full-Newton step feasible interior-point method for convex quadratic programming is presented and analyzed. The idea behind this method is to replace the complementarity condition with a non-negative weight vector and use the algebraic equivalent transformation for the obtained equation. Under the selection of appropriate parameters, the quadratic rate of convergence of the new algorithm is established. In addition, the iteration complexity of the algorithm is obtained. Finally, some numerical results are presented to demonstrate the practical performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Many problems of practical interest can be modeled and solved by using interval-valued fuzzy graph (IVFG) algorithms. An IVFG is a very useful and effective tool for studying various calculations, fields of intelligence, and computer science, such as networking, imaging, and other fields, such as biological sciences. In different applications, they present an appropriate construction means. There were limitations in the definition of fuzzy graphs (FGs), which prompted us to propose a new definition for IVFGs. Some interesting properties related to the new IVFGs are investigated, and enough conditions under which the level graph on IVFGs is equivalent are obtained. Therefore, in this study, we present the properties of a level graph (LG) of an IVFG, and four operations, the Cartesian product (CP), composition (CO), union, and join, are investigated on it. Today, in a treatment system, one of the issues that can be very valuable and important to the quality of service to patients is finding qualified and efficient people in each department, which is not an easy task. But the interval-valued fuzzy graph, as an important fuzzy graph, can help us by considering the ability of each person in the form of intervals of numbers and the effectiveness of each one on the other (according to the relationships between them) in order to find the most worthy people. So, an application of IVFG to find the most effective person in a hospital information system has been introduced. [ABSTRACT FROM AUTHOR]

An important development in geometric algebra in recent years is the new system known as point geometry, which treats points as direct objects of operations and considerably simplifies the process of geometric reasoning. In this paper, we provide a complete formal description of the point geometry theory architecture and give a rigorous and reliable formal verification of the point geometry theory based on the theorem prover Coq. Simultaneously, a series of tactics are also designed to assist in the proof of geometric propositions. Based on the theoretical architecture and proof tactics, a universal and scalable interactive point geometry machine proof system, PointGeo, is built. In this system, any arbitrary point-geometry-solvable geometric statement may be proven, along with readable information about the solution's procedure. Additionally, users may augment the rule base by adding trustworthy rules as needed for certain issues. The implementation of the system expands the library of Coq resources on geometric algebra, which will become a significant research foundation for the fields of geometric algebra, computer science, mathematics education, and other related fields. [ABSTRACT FROM AUTHOR]

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results. [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 S (m , e) be a class of numerical semigroups with multiplicity m and embedding dimension e. We call a graph G S an S (m , e) -graph if there exists a numerical semigroup S ∈ S (m , e) with V (G S) = { x : x ∈ g (S) } and E (G S) = { x y ⇔ x + y ∈ S } , where g (S) denotes the gap set of S. The aim of this article is to discuss the planarity of S (m , e) -graphs for some cases where S is an irreducible numerical semigroup. [ABSTRACT FROM AUTHOR]

The concept of convex ordered hyperrings associated with a strongly regular relation was investigated in this study. In this paper, we first studied hyperatom elements of ordered hyperrings and then investigated characterizations of quotient ordered rings. Is there a strongly regular relation θ on a convex ordered hyperring R for which R / θ is a convex ordered ring? This leads to an ordered ring obtained from an ordered hyperring. [ABSTRACT FROM AUTHOR]

We give some conditions on ordered Γ -semihypergroups under which their interior hyperideal is equal to the hyperideal. In this paper, it is shown that in regular (resp., intraregular, semisimple) ordered Γ -semihypergroups, the hyperideals and the interior hyperideals coincide. To show the importance of these results, some examples and conclusions are provided. [ABSTRACT FROM AUTHOR]

The telescopic numerical semigroups are a subclass of symmetric numerical semigroups widely used in algebraic geometric codes. Suer and Ilhan gave the classification of triply generated telescopic numerical semigroups up to multiplicity 10 and by using this classification they computed some important invariants in terms of the minimal system of generators. In this article, we extend the results of Suer and Ilhan for telescopic numerical semigroups of multiplicities 8 and 12 with embedding dimension four. Furthermore, we compute two important invariants namely the Frobenius number and genus for these classes in terms of the minimal system of generators. [ABSTRACT FROM AUTHOR]

In this study, we concentrate on an important class of ordered hyperstructures with symmetrical hyperoperations, which are called ordered Krasner hyperrings, and discuss strong derivations and homo-derivations. Additionally, we apply our results on nonzero proper hyperideals to the study of derivations of prime ordered hyperrings. This work is a pioneer in studies on structures such as hyperideals and homomorphisms of an ordered hyperring with the help of derivation notation. Finally, we prove some results on 2-torsion-free prime ordered hyperrings by using derivations. We show that if d is a derivation of 2-torsion-free prime hyperring R and the commutator set [ l , d (q) ] is equal to zero for all q in R, then l ∈ Z (R) . Moreover, we prove that if the commutator set (d (l) , q) is equal to zero for all l in R, then (d (R) , q) = 0 . [ABSTRACT FROM AUTHOR]

A total k-rainbow dominating function on a graph G = (V , E) is a function f : V (G) → 2 { 1 , 2 , ... , k } such that (i) ∪ u ∈ N (v) f (u) = { 1 , 2 , ... , k } for every vertex v with f (v) = ∅ , (ii) ∪ u ∈ N (v) f (u) ≠ ∅ for f (v) ≠ ∅ . The weight of a total 2-rainbow dominating function is denoted by ω (f) = ∑ v ∈ V (G) | f (v) | . The total k-rainbow domination number of G is the minimum weight of a total k-rainbow dominating function of G. The minimum total 2-rainbow domination problem (MT2RDP) is to find the total 2-rainbow domination number of the input graph. In this paper, we study the total 2-rainbow domination number of graphs. We prove that the MT2RDP is NP-complete for planar bipartite graphs, chordal bipartite graphs, undirected path graphs and split graphs. Then, a linear-time algorithm is proposed for computing the total k-rainbow domination number of trees. Finally, we study the difference in complexity between MT2RDP and the minimum 2-rainbow domination problem. [ABSTRACT FROM AUTHOR]

Domination in graphs has been extensively studied and adopted in many real life applications. The monitoring electrical power system is a variant of a domination problem called power domination problem. Another variant is the zero forcing problem. Determining minimum cardinality of a power dominating set and zero forcing set in a graph are the power domination problem and zero forcing problem, respectively. Both problems are N P -complete. In this paper, we compute the power domination number and the zero forcing number for fully connected cubic networks. [ABSTRACT FROM AUTHOR]

Predicates and rules are usually enclosed as built-in functions in automated geometry reasoning systems, meaning users cannot add any predicate or rule, thus resulting in a limited reasoning capability of the systems. A method for expanding predicates and rules in automated geometry reasoning systems is, thus, proposed. Specifically, predicate and rule descriptions are transformed to knowledge trees and forests based on formal representations of geometric knowledge, and executable codes are dynamically and automatically generated by using "code templates". Thus, a transformation from controlled natural language descriptions to mechanization algorithms is completed, and finally, the dynamic expansion of predicates and rules in the reasoning system is achieved. Moreover, the method has been implemented in an automated geometry reasoning system for Chinese college entrance examination questions, and the practicality and effectiveness of the method were tested. In conclusion, the enclosed setting, which is a shortcoming of traditional reasoning systems, is avoided, the user-defined dynamic expansion of predicates and rules is realized, the application scope of the reasoning system is extended, and the reasoning capability is improved. [ABSTRACT FROM AUTHOR]

Vague incidence graph (VIG), belonging to the FG family has good capabilities when facing with problems that cannot be expressed by FGs. When an element membership is not clear, neutrality is a good option that can be well-supported by a VIG. The previous definitions limitations in connectivity concept have led us to offer new definitions in VIGs. Hence, in this paper, the VIG and its matrix form are proposed. Vague incidence subgraph (VISG) is defined with several properties. Incidence pairs, paths, and connectivities between pairs in VIGs are introduced. Likewise, different types of strong and cut pair in VIGs are examined with their properties. Universities are one of the most important centers for human education and play an important role in the development of the country. But the point to be very careful is that the employees of a university must do their job in the best possible way. Therefore, we have tried to identify the most effective person in a university according to its performance by presenting an application. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study fundamental properties of weak Γ -hyperfilters on ordered Γ -semihypergroups that is a generalization of Γ -hyperfilters. Also, we investigate the relationship between weak Γ -hyperfilters and prime Γ -hyperideals in ordered Γ -semihypergroups. Finally, we introduce weak m , n - Γ -hyperfilters of ordered Γ -semihypergroups and conclude in paper with arising explicit theorems. [ABSTRACT FROM AUTHOR]

The vague graph (VG), which has recently gained a place in the family of fuzzy graph (FG), has shown good capabilities in the face of problems that cannot be expressed by fuzzy graphs and interval-valued fuzzy graphs. Connectivity index (CI) in graphs is a fundamental issue in fuzzy graph theory that has wide applications in the real world. The previous definitions' limitations in the connectivity of fuzzy graphs directed us to offer new classifications in vague graph. Hence, in this paper, we investigate connectivity index, average connectivity index, and Randic index in vague graphs with several examples. Also, one of the motives of this research is to introduce some special types of vertices such as vague connectivity enhancing vertex, vague connectivity reducing vertex, and vague connectivity neutral vertex with their properties. Finally, an application of connectivity index in the selected town for building hospital is presented. [ABSTRACT FROM AUTHOR]

With the development of network technology, many online educational resource platforms have emerged, and the number of resources on these platforms is increasing dramatically. Compared with the traditional resources, the interactive educational resources (IERs) have hidden interactive information and dynamic content. Only when the users perform interactive operations correctly can they acquire the knowledge conveyed by the resources, which makes it a challenge to help the users understand the general content of resources and find the resources that they are interested in. Therefore, in this paper, we propose a method for generating picture previews of IERs. Firstly, we analyze the file that stores interactive resource information, obtain the interactive elements' information, extract the dependencies, and generate an operation sequence. Secondly, we execute the operation sequence using an automated software tool to mimic the interaction process between the user and the resource and save the screenshots in the process to generate a picture preview of an IER. We apply this method to a dynamic geometry platform, NetPad, and conduct experiments. Experimental results demonstrate that the method can effectively extract interactive semantic information and simulate the interactive process, and the generated picture previews are of great help to the users to understand the content of IERs. [ABSTRACT FROM AUTHOR]

Let Λ be a numerical semigroup and I ⊂ Λ be an irreducible ideal of Λ. The graph G I Λ assigned to an ideal I of Λ is a graph with elements of Λ ∖ I ∗ as vertices, and any two vertices x and y are adjacent if and only if x + y ∈ I. In this work, we give a complete characterization (up to isomorphism) of the graph G I Λ having metric dimension 2. [ABSTRACT FROM AUTHOR]

The bond incident degree (BID) indices can be written as a linear combination of the number of edges x i , j with end vertices of degree i and j. We introduce two transformations, namely, linearizing and unbranching, on catacondensed pentagonal systems and show that BID indices are monotone with respect to these transformations. We derive a general expression for calculating the BID indices of any catacondensed pentagonal system with a given number of pentagons, angular pentagons, and branched pentagons. Finally, we characterize the CPSs for which BID indices assume extremal values and compute their BID indices. [ABSTRACT FROM AUTHOR]

A total k -coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total Δ G + 2 -coloring, where Δ G is the maximum degree of G. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of Δ G ≥ 9 or Δ G ∈ 7,8 with some restrictions has a total Δ G + 1 -coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p , q , ℓ for some p , q , ℓ ∈ 3,4,4 , 3,3,4 . [ABSTRACT FROM AUTHOR]

MicroRNAs (miRNAs) are a class of important non-coding RNAs, which play important roles in tumorigenesis and development by targeting oncogenes or tumor suppressor genes. One miRNA can regulate multiple genes, and one gene can be regulated by multiple miRNAs. To promote the clinical application of miRNAs, two fundamental questions should be answered: what's the regulatory mechanism of a miRNA to a gene, and which miRNAs are important for a specific type of cancer. In this study, we propose a miRNA influence capturing (miRNAInf) to decipher regulation relations of miRNAs on target genes and identify critical miRNAs in cancers in a systematic approach. With the pair-wise miRNA/gene expression profiles data, we consider the assigning problem of a miRNA on target genes and determine the regulatory mechanisms by computing the Pearson correlation coefficient between the expression changes of a miRNA and that of its target gene. Furthermore, we compute the relative local influence strength of a miRNA on its target gene. Finally, integrate the local influence strength and target gene's importance to determine the critical miRNAs involved in specific cancer. Results on breast, liver and prostate cancers show that positive regulations are as common as negative regulations. The top-ranked miRNAs show great potential as therapeutic targets driving cancer to a normal state, and they are demonstrated to be closely related to cancers based on biological functional analysis, drug sensitivity/resistance analysis and survival analysis. This study will be helpful for the discovery of critical miRNAs and development of miRNAs-based clinical therapeutics. [ABSTRACT FROM AUTHOR]

The class of weak pseudoorders and left weak interior hyperideals in ordered hyperstructures is a generalization of pseudoorders and interior hyperideals. In this work, we study the concept of weak pseudoorders and left weak interior hyperideals in ordered hyperstructures and explore some results concerning the new defined concepts for ordered hyperrings and ordered Γ-semihypergroups. In this regards, we intend to concentrate our efforts on the relationship between the left weak interior hyperideal and interior hyperideal of an ordered hyperstructure. A characterization of a regular ordered hyperstructure via a left weak interior hyperideal is given. Finally, we characterize the notion of left weak interior simple ordered hyperstructures in terms of left weak interior hyperideals. [ABSTRACT FROM AUTHOR]

Many problems of practical interest can be modeled and solved by using fuzzy graph (FG) algorithms. In general, fuzzy graph theory has a wide range of application in various fields. Since indeterminate information is an essential real-life problem and is often uncertain, modeling these problems based on FG is highly demanding for an expert. A vague graph (VG) can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems in which fuzzy graphs may not succeed in bringing about satisfactory results. Domination in FGs theory is one of the most widely used concepts in various sciences, including psychology, computer sciences, nervous systems, artificial intelligence, decision-making theory, etc. Many research studies today are trying to find other applications for domination in their field of interest. Hence, in this paper, we introduce different kinds of domination sets, such as the edge dominating set (EDS), the total edge dominating set (TEDS), the global dominating set (GDS), and the restrained dominating set (RDS), in product vague graphs (PVGs) and try to represent the properties of each by giving some examples. The relation between independent edge sets (IESs) and edge covering sets (ECSs) are established. Moreover, we derive the necessary and sufficient conditions for an edge dominating set to be minimal and show when a dominance set can be a global dominance set. Finally, we try to explain the relationship between a restrained dominating set and a restrained independent set with an example. Today, we see that there are still diseases that can only be treated in certain countries because they require a long treatment period with special medical devices. One of these diseases is leukemia, which severely affects the immune system and the body's defenses, making it impossible for the patient to continue living a normal life. Therefore, in this paper, using a dominating set, we try to categorize countries that are in a more favorable position in terms of medical facilities, so that we can transfer the patients to a suitable hospital in the countries better suited in terms of both cost and distance. [ABSTRACT FROM AUTHOR]

Fuzzy graphs (FGs), broadly known as fuzzy incidence graphs (FIGs), have been acknowledged as being an applicable and well-organized tool to epitomize and solve many multifarious real-world problems in which vague data and information are essential. Owing to unpredictable and unspecified information being an integral component in real-life problems that are often uncertain, it is highly challenging for an expert to illustrate those problems through a fuzzy graph. Therefore, resolving the uncertainty accompanying the unpredictable and unspecified information of any real-world problem can be done by applying a vague incidence graph (VIG), based on which the FGs may not engender satisfactory results. Similarly, VIGs are outstandingly practical tools for analyzing different computer science domains such as networking, clustering, and also other issues such as medical sciences, and traffic planning. Dominating sets (DSs) enjoy practical interest in several areas. In wireless networking, DSs are being used to find efficient routes with ad-hoc mobile networks. They have also been employed in document summarization, and in secure systems designs for electrical grids; consequently, in this paper, we extend the concept of the FIG to the VIG, and show some of its important properties. In particular, we discuss the well-known problems of vague incidence dominating set, valid degree, isolated vertex, vague incidence irredundant set and their cardinalities related to the dominating, etc. Finally, a DS application for VIG to properly manage the COVID-19 testing facility is introduced. [ABSTRACT FROM AUTHOR]

A new generation of universal tools and languages for modeling and simulation multi-physical domain applications has emerged and became widely accepted; they generate large-scale systems of differential algebraic equations (DAEs) automatically. Motivated by the characteristics of DAE systems with large dimensions, high index or block structures, we first propose a modified Pantelides' algorithm (MPA) for any high order DAEs based on the Σ matrix, which is similar to Pryce's Σ method. By introducing a vital parameter vector, a modified Pantelides' algorithm with parameters has been presented. It leads to a block Pantelides' algorithm (BPA) naturally which can immediately compute the crucial canonical offsets for whole (coupled) systems with block-triangular form. We illustrate these algorithms by some examples, and preliminary numerical experiments show that the time complexity of BPA can be reduced by at least O (ℓ) compared to the MPA, which is mainly consistent with the results of our analysis. [ABSTRACT FROM AUTHOR]

Fuzzy graph models enjoy the ubiquity of being in natural and human-made structures, namely dynamic process in physical, biological and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems which are often uncertain, it is highly difficult for an expert to model those problems based on a fuzzy graph (FG). Vague graph structure (VGS) can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. Likewise, VGSs are very useful tools for the study of different domains of computer science such as networking, capturing the image, clustering, and also other issues like bioscience, medical science, and traffic plan. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGSs. Operations are conveniently used in many combinatorial applications. In various situations, they present a suitable construction means; therefore, in this research, three new operations on VGSs, namely, maximal product, rejection, residue product were presented, and some results concerning their degrees and total degrees were introduced. Irregularity definitions have been of high significance in the network heterogeneity study, which have implications in networks found across biology, ecology and economy; so special concepts of irregular VGSs with several key properties were explained. Today one of the most important applications of decision making is in medical science for diagnosing the patient's disease. Hence, we recommend an application of VGS in medical diagnosis. [ABSTRACT FROM AUTHOR]

Fuzzy graph models enjoy the ubiquity of being present in nature and man-made structures, such as the dynamic processes in physical, biological, and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems that are often uncertain, for an expert, it is highly difficult to demonstrate those problems through a fuzzy graph. Resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem can be done using a vague graph (VG), with which the fuzzy graphs may not generate satisfactory results. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGs. The objective of this paper is to present certain types of vague graphs (VGs), including strongly irregular (SI), strongly totally irregular (STI), neighborly edge irregular (NEI), and neighborly edge totally irregular vague graphs (NETIVGs), which are introduced for the first time here. Some remarkable properties associated with these new VGs were investigated, and necessary and sufficient conditions under which strongly irregular vague graphs (SIVGs) and highly irregular vague graphs (HIVGs) are equivalent were obtained. The relation among strongly, highly, and neighborly irregular vague graphs was established. A comparative study between NEI and NETIVGs was performed. Different examples are provided to evaluate the validity of the new definitions. A new definition of energy called the Laplacian energy (LE) is presented, and its calculation is shown with some examples. Likewise, we introduce the notions of the adjacency matrix (AM), degree matrix (DM), and Laplacian matrix (LM) of VGs. The lower and upper bounds for the Laplacian energy of a VG are derived. Furthermore, this study discusses the VG energy concept by providing a real-time example. Finally, an application of the proposed concepts is presented to find the most effective person in a hospital. [ABSTRACT FROM AUTHOR]

If G is a graph, its Laplacian is the difference between the diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs G 1 and G 2 is a graph G = G 1 ⊙ u v G 2 with V (G) = V (G 1) ∪ V (G 2) and E (G) = E (G 1) ∪ E (G 2) ∪ { e = u v } where u ∈ V (G 1) and v ∈ V (G 2) . In this paper, we study some structural conditions ensuring the presence of 2 in the Laplacian spectrum of bicyclic graphs of type G 1 ⊙ u v G 2 . We also provide a condition under which a bicyclic graph with a perfect matching has a Laplacian eigenvalue 2. Moreover, we characterize the broken sun graphs and the one-edge connection of two broken sun graphs by their Laplacian eigenvalue 2. [ABSTRACT FROM AUTHOR]