a b s t r a c tThe Wiener index is the sum of distances between all vertex pairs in a connected graph. This notion was motivated by various mathematical properties and chemical applications. In this paper we introduce four new operations on graphs and study the Wiener indices of the resulting graphs.
For a finite group G different from a cyclic group of prime power order, we introduce an undirected simple graph D(G) whose vertices are the proper subgroups of G which are not contained in the Frattini subgroup of G and two vertices H and K are joined by an edge if and only if G hHY Ki. In this paper we study D(G) and show that it is connected and determine the clique and chromatic number of D(G) and obtain bounds for its diameter and girth. We classify finite groups with complete graphs and also classify finite groups with domination number 1. Also we show that if the independence number of the graph D(G) is at most 7, then G is solvable.
A digraph $\Gamma$ is called $n$-Cayley digraph over a group $G$, if there exists a semiregular subgroup $R_G$ of Aut$(\Gamma)$ isomorphic to $G$ with $n$ orbits. In this paper, we represent the adjacency matrix of $\Gamma$ as a diagonal block matrix in terms of irreducible representations of $G$ and determine its characteristic polynomial. As corollaries of this result we find: the spectrum of semi-Cayley graphs over abelian groups, a relation between the characteristic polynomial of an $n$-Cayley graph and its complement, and the spectrum of Calye graphs over groups with cyclic subgroups. Finally we determine the eigenspace of $n$-Cayley digraphs and their main eigenvalues.
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