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We investigate signed graphs with just 2 or 3 distinct eigenvalues, mostly in the context of vertex-deleted subgraphs, the join of two signed graphs or association schemes. [ABSTRACT FROM AUTHOR]

We consider the notion of χ -completeness of a locally finite graph and we extend this notion to the weighted magnetic graph. Moreover, we establish a link between the magnetic adjacency matrix on line graph and the magnetic discrete Laplacian on 1-forms. [ABSTRACT FROM AUTHOR]

For a given complex square matrix A with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first, we derive bounds for the second largest and the smallest eigenvalues of adjacency matrices of fc-regular graphs. Then we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest and the largest eigenvalues of the Laplacian matrices of graphs. The sharpness of these bounds is verified by examples. [ABSTRACT FROM AUTHOR]

The anti-Gallai graph Δ (G) of a graph G, has the edges of G as its vertices and two distinct edges are adjacent in Δ (G) if they lie on a triangle in G. In this paper we characterize the graphs whose anti-Gallai graph has only one or two eigenvalues and study the adjacency spectrum of the anti-Gallai graph of (K 4 , H) -free graphs, where H is the Hajos graph and graphs in which two diamonds share at most one vertex. We also find expressions for the adjacency spectrum of anti-Gallai graph of graphs after applying certain binary operations. [ABSTRACT FROM AUTHOR]

An eigenvalue of a real symmetric matrix is called main if there is an associated eigenvector not orthogonal to the all-1 vector j. Main eigenvalues are frequently considered in the framework of simple undirected graphs. In this study we generalize some results and then apply them to signed graphs. [ABSTRACT FROM AUTHOR]

Connected planar graphs are of interest to a variety of scholars. Being able to simulate a database of such graphs with selected properties would support specific types of inference for spatial analysis and other network-based disciplines. This paper presents a simple, efficient, and flexible connected planar graph generator for this purpose. It also summarizes a comparison between an empirical set of specimen graphs and their simulated counterpart set, and establishes evidence for positing a conjecture about the principal eigenvalue of connected planar graphs. Finally, it summarizes a probability assessment of the algorithm’s outcomes as well as a comparison between the new algorithm and selected existing planar graph generators. [ABSTRACT FROM AUTHOR]

How to mine many interesting subgraphs in uncertain graph has become an important research field in data mining. In this paper, a novel algorithm Uncertain Maximal Frequent Subgraph Mining Algorithm Based on Adjacency Matrix and Weight (UMFGAMW) is proposed. The definition of the adjacency matrix and the standard matrix coding for uncertain graph are presented. The correspondence between the adjacency matrix and uncertain graph is established. A new vertex ordering policy for computing the standard coding of uncertain graph adjacency matrix is designed. The complexity of uncertain graph standard coding is reduced, and the matching speed of uncertain subgraph standard coding is improved. The definition of the weight of uncertain graph and the mean weight of uncertain edge is proposed. The importance of the uncertain subgraphs that meet the minimum support threshold in the graph dataset is fully considered. Finally, a depth-first search weighted uncertain maximal frequent subgraph mining algorithm is discussed. According to the limiting condition of the uncertain maximum frequent subgraph and weighed uncertain edge, the number of mining results is reduced effectively. Experimental results demonstrate that the UMFGAMW algorithm has higher efficiency and better scalability. [ABSTRACT FROM AUTHOR]

In this paper, a tolerance representation model for generating tolerance specification schemes and corresponding tolerance zones is proposed to meet the requirement of the representation of tolerance information semantics. This model is hierarchically organized and consists of five layers. They are component, geometric feature, variational geometric constraint, tolerance specification scheme, and tolerance zone layers. The mating relations between components in the component layer, the mating relations between geometric features in the geometric feature layer, the variational geometric constraints between geometric features in the variational geometric constraint layer, the tolerance specification schemes of component in the tolerance specification scheme layer, and the tolerance zones of tolerance specification schemes in the tolerance zone layer are formally defined by one or more adjacency matrices, respectively. Based on the model, a method for generating tolerance specification schemes for component and their resultant tolerance zones is designed. This method shows how to adopt a top-down strategy to carry out tolerance specification for an arbitrary assembly designed in a CAD system. The paper also provides a practical example to illustrate how the method works. [ABSTRACT FROM AUTHOR]

In this paper, we have designed a new algorithm for calculating the exact value of the chromatic number of a graph and dubbed it as R-coloring algorithm. Using this R-coloring algorithm, we have proved some results. We have discussed some application in different fields of our life by using this new algorithm. [ABSTRACT FROM AUTHOR]

A graph is nonsingular if its adjacency matrix A( G) is nonsingular. The inverse of a nonsingular graph G is a graph whose adjacency matrix is similar to A( G) via a particular type of similarity. Let H denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in H which possess unicyclic inverses. We present a characterization of unicyclic graphs in H which possess bicyclic inverses. [ABSTRACT FROM AUTHOR]

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to $$-2$$ , or 0, and determine which of these graphs are determined by their adjacency spectrum. [ABSTRACT FROM AUTHOR]

A Hamiltonian cycle in a graph is a cycle that visits each node/vertex exactly once. A graph containing a Hamiltonian cycle is called a Hamiltonian graph. There have been several researches to find the number of Hamiltonian cycles of a Hamilton graph. As the number of vertices and edges grow, it becomes very difficult to keep track of all the different ways through which the vertices are connected. Hence, analysis of large graphs can be efficiently done with the assistance of a computer system that interprets graphs as matrices. And, of course, a good and well written algorithm will expedite the analysis even faster. The most convenient way to quickly test whether there is an edge between two vertices is to represent graphs using adjacent matrices. In this paper, a new algorithm is proposed to find fuzzy Hamiltonian cycle using adjacency matrix and the degree of the vertices of a fuzzy graph. A fuzzy graph structure is also modeled to illustrate the proposed algorithms with the selected air network of Indigo airlines. [ABSTRACT FROM AUTHOR]

Sudoku is a popular combinatorial puzzle. We give a brief overview of some mathematical features of a Sudoku square. Then we focus on interpreting Sudoku squares as experimental designs in order to meet a practical need. [ABSTRACT FROM AUTHOR]

In continuation of the results obtained in [3] for the realization of the product of adjacency matrices under usnal matrix multiplication, this article presents some interesting characterizations and properties of the graphs for which the product of adjacency matrices under modulo-2 is graphical. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to show an application of Graph Theory to classifying Lie algebras over finite fields. It is rooted in the representation of each Lie algebra by a certain pseudo-graph. As partial results, it is deduced that there exist, up to isomorphism, four, six, fourteen and thirty-four 2-, 3-, 4-, and 5-dimensional algebras of the studied family, respectively, over the field ℤ/2ℤ. Over ℤ/3ℤ, eight and twenty-two 2- and 3-dimensional Lie algebras, respectively, are also found. Finally, some ideas for future research are presented. [ABSTRACT FROM AUTHOR]

We look for the maximum order m( r) of the adjacency matrix A of a graph G with a fixed rank r, provided A has no repeated rows or all-zero row. Akbari, Cameron and Khosrovshahi conjecture that m( r) = 2 − 2 if r is even, and m( r) = 5 · 2 − 2 if r is odd. We prove the conjecture and characterize G in the case that G contains an induced subgraph $${\frac{r}{2}K_2}$$ or $${\frac{r-3}{2}K_2+K_3}$$. [ABSTRACT FROM AUTHOR]

We study spectral properties of the Manhattan products of the path graphs and show the concentration of zero-eigenvelues. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph of order n with degree sequence ( d, d, ..., d). Denote ( t) = Σ d, ( m) = ( t)/ d and ( N) = Σ ( t), where α is a real number. Denote by λ( G) and µ( G) the spectral radius of the adjacency matrix and the Laplacian matrix of G, respectively. In this paper, we present some upper and lower bounds of λ( G) and µ( G) in terms of ( t), ( m) and ( N). Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results. [ABSTRACT FROM AUTHOR]

A multicone graph is defined to be the join of a clique and a regular graph. Based on Zhou and Cho's result [ B. Zhou, H.H. Cho, Remarks on spectral radius and Laplacian eigenvalues of a graph, Czech. Math. J. 55 ( 130) (2005), 781-790], the spectral characterization of multicone graphs is investigated. Particularly, we determine a necessary and sufficient condition for two multicone graphs to be cospectral graphs and investigate the structures of graphs cospectral to a multicone graph. Additionally, lower and upper bounds for the largest eigenvalue of a multicone graph are given. [ABSTRACT FROM AUTHOR]

The Hoffman-Singleton graph is the unique strongly regular graph with parameters (50, 7, 0, 1). A well-known hypothesis states that a distance-regular graph in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph has intersection array {50, 42, 1; 1, 2, 50} or {50, 42, 9; 1, 2, 42}. In the present paper, we prove this hypothesis under the assumption that a distance-regular graph is a Terwilliger graph and the graph diameter is at most 5. [ABSTRACT FROM AUTHOR]

Possible orders and fixed-point subgraphs of automorphisms of a generalized octagon of order (2, 4) are found. The vertex-symmetric generalized octagon of order (2, 4) is proved to be classical. [ABSTRACT FROM AUTHOR]

This paper presents a new MSDI (mold surface data integration) method for presenting ruled mold-surface data for CAPP applications. The method makes use of adjacency and feature-data matrices to establish a framework of feature-based data. The MSDI method enables the storage, retrieval and operations of geometries, topologies, and machining information of a given ruled mold surface. The matrices of adjacency and feature-data effectively represent the relations. Boolean algebra and group technology (GT) methods are introduced to generate the desired process plans. A sample mold made of ruled surfaces is made and used to show the effectiveness of the proposed data representation and the integrated mold surface processing method. A compact computer program is implemented to enhance the speed and accuracy of the MSDI method. The numerical results indicate that the mold surface decomposition and integration procedure presented in this paper are systematic and effective. The paper provides a novel modular mold surface modeling tool and procedure that can be further extended to accommodate more features appearing in complicated ruled mold surfaces. [ABSTRACT FROM AUTHOR]

It is proved that any set of finite pairwise nonisomorphic weakly cospectral connected pseudographs, as well as any set of pseudodigraphs satisfying various additional conditions, is finite. [ABSTRACT FROM AUTHOR]

A simple pictorial algorithm for factorisation of symmetric chemical graphs (weighted and unweighted) leading to simultaneous determination of their eigenvalues and eigenvectors has been devised. The method does not require group-theoretical techniques (viz. identification of the point group of the species under study, formation of symmetryadopted linear combinations using character tables etc.). It requires consideration of only one symmetry element, e.g., a reflection plane and is based on elementary row and column operations which keep the secular determinant of the adjacency matrix unchanged (except possibly for a multiplicative constant). [ABSTRACT FROM AUTHOR]

Let (G,∗) be a finite group and S={x∈G|x≠x−1} be a subset of G containing its non-self invertible elements. The inverse graph of G denoted by Γ(G) is a graph whose set of vertices coincides with G such that two distinct vertices x and y are adjacent if either x∗y∈S or y∗x∈S. In this paper, we study the energy of the dihedral and symmetric groups, we show that if G is a finite non-abelian group with exactly two non-self invertible elements, then the associated inverse graph Γ(G) is never a complete bipartite graph. Furthermore, we establish the isomorphism between the inverse graphs of a subgroup Dp of the dihedral group Dn of order 2p and subgroup Sk of the symmetric groups Sn of order k! such that 2 p = n ! (p , n , k ≥ 3 and p , n , k ∈ ℤ +). [ABSTRACT FROM AUTHOR]