An antipodal graph D of diameter d has the property that each vertex v has a unique (antipodal) vertex v at distance d from v in D. We show that any such D has circuits of length Id passing through antipodal pairs of vertices. The identification of antipodal vertex-pairs in D produces a quotient graph G with a double cover projection morphism p : D-+G. Using the two-fold quotient map of surface...

The symmetry properties of mathematical structures are often important for understanding these structures. Graphs sometimes have a large group of symmetries, especially when they have an algebraic construction such as the Cayley graphs. These graphs are constructed from abstract groups and are vertex-transitive and this is the reason for their symmetric appearance. Some Cayley graphs have even ...

We find a natural construction of a large class of symmetric graphs from pointand block-transitive 1-designs. The graphs in this class can be characterized as G-symmetric graphs whose vertex sets admit a G-invariant partition B of block size at least 3 such that, for any two blocks B,C of B, either there is no edge between B and C , or there exists only one vertex in B not adjacent to any verte...

emph{The (undirected) power graph on the conjugacy classes} $mathcal{P_C}(G)$ of a group $G$ is a simple graph in which the vertices are the conjugacy classes of $G$ and two distinct vertices $C$ and $C'$ are adjacent in $mathcal{P_C}(G)$ if one is a subset of a power of the other. In this paper, we describe groups whose associated graphs are $k$-regular for $k=5,6$.

A finite graph Γ is G-symmetric if it admits G as a group of automorphisms acting transitively on V (Γ) and transitively on the set of ordered pairs of adjacent vertices of Γ. If V (Γ) admits a nontrivial G-invariant partition B such that for blocks B,C ∈ B adjacent in the quotient graph ΓB relative to B, exactly one vertex of B has no neighbour in C, then we say that Γ is an almost multicover ...

Reed and Seymour [1998] asked whether every graph has a partition into induced connected subgraphs such that each part is bipartite and the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger’s Conjecture. We prove that the answer is ‘no’. In fact, we show that the answer is still ‘no’ for several relaxations of the question.

Let be a G-symmetric graph admitting a nontrivial G-invariant partition B. Let B be the quotient graph of with respect to B. For each block B ∈ B, the setwise stabiliser G B of B in G induces natural actions on B and on the neighbourhood B(B) of B in B . Let G(B) and G[B] be respectively the kernels of these actions. In this paper we study certain “local actions” induced by G(B) and G[B], such ...

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...

Generalized Pareto distribution play an important role in reliability, extreme value theory, and other branches of applied probability and statistics. This family of distributions includes exponential distribution, Pareto distribution, and Power distribution. In this paper, we established exact expressions and recurrence relations satisfied by the quotient moments of generalized order statistic...

We show that the tangle space of a graph, which compactifies it, is quotient its Stone-Čech remainder obtained by contracting connected components.