1. Introduction. A graph G consists of a non-null set V of objects called vertices together with a set E of objects called edges, the two sets having no common element. With each edge there are associated just two vertices, called its ends. Two or more edges may have the same pair of ends. G is finite if both F and E are finite, and infinite otherwise. The degree d G {a) of a vertex a of G is t...

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing K4 or K2,3 as a minor is Hamiltonian if and only if it is 2-connected. We show furth...

If E is a not-necessarily row-finite graph, such that each vertex of E emits at most countably many edges, then a desingularization F of E can be constructed as described in [3] or [7]. The desingularization process has been effectively used to establish various characteristics of the Leavitt path algebras of not-necessarily row-finite graphs. Such a desingularization F of E has the properties ...

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...

Any hereditarily finite set S can be represented as a finite pointed graph –dubbed membership graph– whose nodes denote elements of the transitive closure of {S} and whose edges model the membership relation. Membership graphs must be hyper-extensional –nodes are pairwise not bisimilar– and bisimilar nodes represent the same hereditarily finite set. It is worth to notice that the removal of eve...

Let $G$ be a finite group and $pi_{e}(G)$ be the set of orders of all elements in $G$. The set $pi_{e}(G)$ determines the prime graph (or Grunberg-Kegel graph) $Gamma(G)$ whose vertex set is $pi(G)$, the set of primes dividing the order of $G$, and two vertices $p$ and $q$ are adjacent if and only if $pqinpi_{e}(G)$. The degree $deg(p)$ of a vertex $pin pi(G)$, is the number of edges incident...