This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$, there is an embedding of $(G,d)$ (possibly after a rescaling of $d$) on a surface of genus $g$. Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$, where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.

Let $R$ be a commutative ring with non-zero identity. In this paper, we introduce the weakly nilpotent graph of a commutative ring. The weakly nilpotent graph of $R$ denoted by ${{\Gamma }_{w}}(R)$ is a graph with the vertex set ${{R}^{\star }}$ and two vertices $x$ and $y$ are adjacent if and only if $x\,y\in N{{(R)}^{\star }}$, where ${{R}^{\star }}=R\backslash \{0\}$ and $N{{(R)}^{\star }}$ is the set of all non-zero nilpotent elements of $R$. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if ${{\Gamma }_{w}}(R)$ is a forest, then ${{\Gamma }_{w}}(R)$ is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of ${{\Gamma }_{w}}(R)$. Among other results, we show that for an Artinian ring $R$, ${{\Gamma }_{w}}(R)$ is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam $\overline{({{\Gamma }_{w}}(R))}$. Finally, we characterize all commutative rings $R$ for which $\overline{({{\Gamma }_{w}}(R))}$ is a cycle, where $\overline{({{\Gamma }_{w}}(R))}$ is the complement of the weakly nilpotent graph of $R$.

Let $R$ be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on $R$, $\Gamma \left( R \right)$, with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra\,+\,Rb\,=\,R$. In this paper, we consider a subgraph ${{\Gamma }_{2}}\left( R \right)$ of $\Gamma \left( R \right)$ that consists of non-unit elements. We investigate the behavior of ${{\Gamma }_{2}}\left( R \right)$ and ${{\Gamma }_{2}}\left( R \right)\backslash \text{J}\left( R \right)$, where $\text{J}\left( R \right)$ is the Jacobson radical of $R$. We associate the ring properties of $R$, the graph properties of ${{\Gamma }_{2}}\left( R \right)$, and the topological properties of $\text{Max}\left( R \right)$. Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.

In this paper, we determine when $\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph ${\Gamma }_{I} (L)$ with respect to a semiprime ideal $I$ of a lattice $L$, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck’s conjecture is proved for ${\Gamma }_{I} (L)$ when $\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $.

Let $R$ be a ring and $Z(R)$ be the set of all zero-divisors of $R$. The total graph of $R$, denoted by $T(\Gamma (R))$ is a graph with all elements of $R$ as vertices, and two distinct vertices $x, y\in R$ are adjacent if and only if $x+ y\in Z(R)$. Let the regular graph of $R$, $\mathrm{Reg} (\Gamma (R))$, be the induced subgraph of $T(\Gamma (R))$ on the regular elements of $R$. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set $\{ 3, 4, \infty \} $. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if $R$ is a reduced left Noetherian ring and $2\not\in Z(R)$, then the chromatic number and the clique number of $\mathrm{Reg} (\Gamma (R))$ are the same and they are ${2}^{r} $, where $r$ is the number of minimal prime ideals of $R$. Among other results, we show that if $R$ is a semiprime left Noetherian ring and $\mathrm{Reg} (R)$ is finite, then $R$ is finite.

Let $C$ and $D$ be digraphs. A mapping $f:V\left( D \right)\to V\left( C \right)$ is a $C$-colouring if for every arc $uv$ of $D$, either $f\left( u \right)f\left( v \right)$ is an arc of $C$ or $f\left( u \right)=f\left( v \right)$, and the preimage of every vertex of $C$ induces an acyclic subdigraph in $D$. We say that $D$ is $C$-colourable if it admits a $C$-colouring and that $D$ is uniquely $C$-colourable if it is surjectively $C$-colourable and any two $C$-colourings of $D$ differ by an automorphism of $C$. We prove that if a digraph $D$ is not $C$-colourable, then there exist digraphs of arbitrarily large girth that are $D$-colourable but not $C$-colourable. Moreover, for every digraph $D$ that is uniquely $D$-colourable, there exists a uniquely $D$-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number $r\ge 1$, there are uniquely circularly $r$-colourable digraphs with arbitrarily large girth.

The bipartite divisor graph B(X), for a set Xof positive integers, and some of its properties have recently been studied. We construct the bipartite divisor graph for the product of subsets of positive integers and investigate some of its properties. We also give some applications in group theory.

The tension applied to the girth is usually based on an individual's experience rather than by scientific measurement or procedure. The equine thorax is a dynamic structure, and therefore the actual readings of girth tensions at rest and during exercise (actual tension) are likely to vary from the tension to which the girth was intended to be tightened at rest while standing (intended tension). This study was undertaken to determine the variability of girth tensions at rest and during exercise. A total of 19 Hanoverian horses were lunged on a 20 m circle in walk, trot and canter on both reins. In a randomized design, each horse was exposed to intended tensions 6, 10, 14 and 18 kg (saddle and girth at appropriate intended tensions). Girth tension was measured and recorded continuously using an in-line load cell. Intended girth tensions were not significantly different with mean actual girth tensions while standing. Actual girth tensions increased significantly (P < 0.001) between walk, trot and canter at all tensions except rest to trot at tension 6 kg, where the significance level was P < 0.01. Actual girth tension was significantly higher (P < 0.001) on the left rein at tension 14 kg in walk and trot, and at tensions 6, 10 and 14 kg during canter, and there was an overall trend for higher actual girth tensions on the left rein for the other tensions. As the thorax is a dynamic structure, girth tension variation could be due to multiple factors such as respiration, breath holding, muscular contraction, back flexion and extension, speed, gait and vertical acceleration of the saddle. Girth tension is a relatively new area of research, and as there are many opportunities for further research, a better understanding of the impact the girth has on the horse could help to improve performance and welfare.

On étudie à l'aide du système AutoGraphiX 2 (AGX 2) des relations de la forme \[ \underline{b}_{n} \, \le \, g\,\oplus \,i\, \le \, \overline{b}_{n}\] où g désigne la maille d'un graphe G=(V, E), i un autre invariant parmi la distance moyenne $\overline{l}$ , l'index λ1, l'indice de Randić R et le nombre de domination β, $\oplus$ désigne l'une des opérations+, -, ×, /, $\underline{b}_{n}$ et $\overline{b}_{n}$ des fonctions de l'ordre n du graphe qui bornent l'expression $g\oplus i$ et sont atteintes pour tout n (sauf éventuellement de très petites valeurs du fait des effets de bord). Les résultats prouvés ou discutés ci-dessous ont déjà été présentés, sousforme de conjectures, dans un article précédent paru dans RAIRO Recherche Opérationnelle [RAIRO Oper. Res. 39 (2005) 275–293].

Some simplifications of Schäffer's girth and perimeter of the unit spheres are introduced. Their general properties are discussed, and they are used to study the lp, Lp spaces, uniformly nonsquare spaces, and their isomorphic classes.

Lower bounds are given for the independence ratio in graphs satisfying certain girth and maximum degree requirements. In particular, the independence ratio of a graph with maximum degree Δ and girth at least six is at least (2Δ − 1)/(Δ2 + 2Δ − 1). Sharper bounds are given for cubic graphs.