It is well-known that any graph has all real eigenvalues and a graph is bipartite if and only if its spectrum is symmetric with respect to the origin. We are interested in finding whether the permanental roots of a bipartite graph G have symmetric property as the spectrum of G. In this note, we show that the permanental roots of bipartite graphs are symmetric with respect to the real and imagin...

We prove that the underlying graphs of p-gonal r-valent orientably regular maps are arc-transitive but non-Cayley if r 3 and p is a prime greater than r(r ? 1).

We establish natural bijections between three different classes of combinatorial objects; namely certain families of locally 2–arc transitive graphs, partial linear spaces, and homogeneous factorisations of arc-transitive graphs. Moreover, the bijections intertwine the actions of the relevant automorphism groups. Thus constructions in any of these areas provide examples for the others.

A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Letn be a product of three primes. The problemon the classification of the tetravalent half-arc-transitive graphs of order n has been considered by Xu (1992), Feng et al. (2007) and Wang and Feng (2010), and it was solved for the cases where n is a prime cube or twice a produ...

We consider in this paper only non-directed graphs without multiple edges and without loops . The number of vertices of a graph G will be called its order, and will be denoted by N(G) . We shall call such a graph symmetric, if there exists a non-identical permutation of its vertices, which leaves the graph invariant. By other words a graph is called symmetric if the group of its automorphisms h...

This paper presents a novel symmetric graph regularization framework for pairwise constraint propagation. We first decompose the challenging problem of pairwise constraint propagation into a series of two-class label propagation subproblems and then deal with these subproblems by quadratic optimization with symmetric graph regularization. More importantly, we clearly show that pairwise constrai...

Let $G$ be a finite group and $Gamma(G)$ the prime graph of $G$‎. ‎Recently people have been using prime graphs to study simple groups‎. ‎Naturally we pose a question‎: ‎can we use prime graphs to study almost‎ ‎simple groups or non-simple groups? In this paper some results in‎ ‎this respect are obtained and as follows‎: ‎$Gcong S_p$ if and only‎ ‎if $|G|=|S_p|$ and $Gamma(G)=Gamma(S_p)$‎, ‎whe...

The symmetric m-th power of a graph is the graph whose vertices are m-subsets of vertices and in which two m-subsets are adjacent if and only if their symmetric difference is an edge of the original graph. It was conjectured that there exists a fixed m such that any two graphs are isomorphic if and only if their m-th symmetric powers are cospectral. In this paper we show that given a positive i...

Networks play an important role in electrical and electronic engineering. It depends on what area of engineering, for example, there is a lot more abstract mathematics communication theory signal processing networking, etc. involve nodes communicating with each other. Graph has found considerable use this area. In paper, we introduce some new such as Graph-PW, Network Symmetric Digraph-PW, Chan...