Let G be a group acting on a finite set Ω. Then G acts on Ω × Ω by its entry-wise action and its orbits form the basis relations of a coherent configuration (or shortly scheme). Our concern is to consider what follows from the assumption that the number of orbits of G on Ω i × Ω j is constant whenever Ω i and Ω j are orbits of G on Ω. One can conclude from the assumption that the actions of G on Ω i 's have the same permutation character and are not necessarily equivalent. From this viewpoint one may ask how many inequivalent actions of a given group with the same permutation character there exist. In this article we will approach to this question by a purely combinatorial method in terms of schemes and investigate the following topics: (i) balanced schemes and their central primitive idempotents, (ii) characterization of reduced balanced schemes.
The distance d(u, v) between vertices u and v of a connected graph G is equal to the number of edges in a minimal path connecting them. The transmission of a vertex v is defined by σ(v) = u∈V (G) d(v, u). A topological index is said to be a transmission-based topological index (TT index) if it includes the transmissions σ(u) of vertices of G. Because σ(u) can be derived from the distance matrix of G, it follows that transmission-based topological indices form a subset of distance-based topological indices. So far, relatively limited attention has been paid to TT indices, and very little systematic studies have been done. In this paper our aim was i) to define various types of transmission-based topological indices ii) establish lower and upper bounds for them, and iii) determine a family of graphs for which these bounds are best possible. Additionally, it has been shown in examples that using a group theoretical approach the transmission-based topological indices can be easily computed for a particular set of regular, vertex-transitive, and edge-transitive graphs. Finally, it is demonstrated that there exist TT indices which can be successfully applied to predict various physicochemical properties of different organic compounds. Some of them give better results and have a better discriminatory power than the most popular degree-based and distance-based indices (Randić, Wiener, Balaban indices).
Signless Laplacian determinations of some graphs with independent edgesLet G be a simple undirected graph. Then the signless Laplacian matrix of G is defined as D G + A G in which D G and A G denote the degree matrix and the adjacency matrix of G, respectively. The graph G is said to be determined by its signless Laplacian spectrum (DQS, for short), if any graph having the same signless Laplacian spectrum as G is isomorphic to G. We show that G rK 2 is determined by its signless Laplacian spectra under certain conditions, where r and K 2 denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained.
Abstract:The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G , which in turn is equal to the sum of the singular values of the adjacency matrix of G. Let X, Y and Z be matrices, such that X +Y = Z. The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y . This theorem is applied in the theory of graph energy, resulting in several new inequalities.
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