12 research outputs found
Bounds for the Generalized Distance Eigenvalues of a Graph
Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D(G) be the distance matrix, DL(G) be the distance Laplacian, DQ(G) be the distance signless Laplacian, and Tr(G) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by Dα(G) the generalized distance matrix, i.e., Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue
On Some Aspects of the Generalized Petersen Graph
Let be a positive integer and let . The generalized Petersen graph GP(p,k) has its vertex and edge set as and . In this paper we probe its spectrum and determine the Estrada index, Laplacian Estrada index, signless Laplacian Estrada index, normalized Laplacian Estrada index, and energy of a graph. While obtaining some interesting results, we also provide relevant background and problems
On the Estrada index of unicyclic and bicyclic signed graphs
Let be a signed graph of order with eigenvalues
We define the Estrada index of a signed graph
as . We characterize the signed
unicyclic graphs with the maximum Estrada index. The signed graph is
said to have the pairing property if is an eigenvalue whenever is
an eigenvalue of and both and have the same
multiplicities. If denotes the set of all unbalanced
graphs on vertices and edges with the pairing property, we determine
the signed graphs having the maximum Estrada index in ,
when and . Finally, we find the signed graphs among all unbalanced
complete bipartite signed graphs having the maximum Estrada index.Comment: 16 pages, 2 figure
On the Laplacian and Signless Laplacian Characteristic Polynomials of a Digraph
Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L(D)=Deg+(D)−A(D) and Q(D)=Deg+(D)+A(D), where A(D) represents the adjacency matrix and Deg+(D) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D. We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D. We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations
Laplacian energy of graphs and digraphs.
Spectral graph theory (Algebraic graph theory) which emerged in 1950s and 1960s is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues and eigenvectors of matrices associated to the graph. The major source of research in spectral graph theory has been the study of relationship between the structural and spectral properties of graphs. Another source has research in quantum chemistry. Just as astronomers study stellar spectra to determine the make-up of distant stars, one of the main goals in spectral graph theory is to deduce the principal properties and structure of a graph from its graph spectrum (or from a short list of easily computable invariants). The spectral approach for general graphs is a step in this direction.Digital copy of Thesis.University of Kashmir
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group