The girth of a finitely generated group G is defined to be the supremum of the girth of its Cayley graphs. Let G be a finitely generated subgroup of the mapping class group ModΣ, where Σ is an orientable closed surface with a finite number of punctures and with a finite number of components. We show that G is either a non-cyclic group with infinite girth or a virtually free-abelian group; these...

For a given graph G, its line denoted by L(G) is whose vertex set V (L(G)) = E(G) and {e1, e2} ∈ E(L(G)) if e1 e2 are incident to common in G. Let R be finite commutative ring with nonzero identity G(R) denotes the unit associated R. In this manuscript, we have studied L(G(R)) of course investigation, several basic properties, viz., diameter, girth, clique, chromatic number been determined. Fur...

We investigate the basic interplay between the small k-valent vertex-transitive graphs of girth g and the (k, g)-cages, the smallest k-valent graphs of girth g. We prove the existence of k-valent Cayley graphs of girth g for every pair of parameters k ≥ 2 and g ≥ 3, improve the lower bounds on the order of the smallest (k, g) vertex-transitive graphs for certain families with prime power girth,...

Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number. We investigate here homomorphisms from orienta-tions of undirected planar graphs to graphs (not necessarily planar) containing as few digons as possible. We relate the existence of such homomorphisms to girth and it appears that these questions remain interesting even under large girth assumption ...

The well-known Moore bound M(k, g) serves as a universal lower bound for the order of k-regular graphs of girth g. The excess e of a k-regular graph G of girth g and order n is the difference between its order n and the corresponding Moore bound, e = n −M(k, g). We find infinite families of parameters (k, g), g > 6 and even, for which we show that the excess of any k-regular graph of girth g is...

Girth weld cracking is one of the main failure modes in oil and gas pipelines; girth weld cracking inspection has great economic and social significance for the intrinsic safety of pipelines. This paper introduces the typical girth weld defects of oil and gas pipelines and the common nondestructive testing methods, and systematically generalizes the progress in the studies on technical principl...

We prove that every graph of minimum degree at least r and girth at least 186 contains a subdivision of Krþ1 and that for r5435 a girth of at least 15 suffices. This implies that the conjecture of Haj ! os that every graph of chromatic number at least r contains a subdivision of Kr (which is false in general) is true for graphs of girth at least 186 (or 15 if r5436). More generally, we show tha...

Let G = (V,E) be a graph of order p and size q. It is known that if G is super edge-magic graph then q ≤ 2p− 3. Furthermore, if G is super edge-magic and q = 2p− 3, then the girth of G is 3. It is also known that if the girth of G is at least 4 and G is super edge-magic then q ≤ 2p − 5. In this paper we show that there are infinitely many graphs which are super edge-magic, have girth 5, and q =...

Installation of new pipelines is predicted to grow at a rapid rate over the next twenty years, due in part to the increase use worldwide of combined cycle power generation plant using natural gas a fuel. The need to construct large diameter pipelines over long distances has led to an increased demand to improve the productivity of pipeline girth welding. Many novel techniques have been tried in...