A convex body R in the hyperbolic plane is called reduced if any convex body $K\subset R$ has a smaller minimal width than R. We answer a few of Lassak’s questions about ordinary reduced polygons regarding its perimeter, diameter, and circumradius, and we also obtain a hyperbolic extension of a result of Fabińska.

We give a simplified version of the proofs that, outside of their isolated vertices, the complement of the enhanced power graph and of the power graph are connected and have diameter at most $3$.

A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. Harary and Thomassen [‘Anticritical graphs’, Math. Proc. Cambridge Philos. Soc. 79(1) (1976), 11–18] proved that the radius r and diameter d of any radially maximal graph satisfy $r\le d\le 2r-2.$ Dutton et al. [‘Changing and unchanging of the radius of a graph’, Linear Algebra Appl. 217 (1995), 67–82] rediscovered this result with a different proof and conjectured that the converse is true, that is, if r and d are positive integers satisfying $r\le d\le 2r-2,$ then there exists a radially maximal graph with radius r and diameter $d.$ We prove this conjecture and a little more.

In this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.

In this work, we use rigorous probabilistic methods to study the asymptotic degree distribution, clustering coefficient, and diameter of geographical attachment networks. As a type of small-world network model, these networks were first proposed in the physical literature, where they were analyzed only with heuristic arguments and computational simulations.

Let $F$ be an algebraically closed field of characteristic $0$ and let $\operatorname{sp}(2l,F)$ be the rank $l$ symplectic algebra of all $2l\times 2l$ matrices $x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over $F$, where $A^{t}$ is the transpose of $A$ and $B,C$ are symmetric matrices of order $l$. The commuting graph $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of $\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in $\operatorname{sp}(2l,F)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$. We prove that the diameter of $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is $4$ when $l>2$.

We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes $P$ and $Q$, we show that $P$ can be written as a Minkowski sum with a summand homothetic to $Q$ if and only if $P$ has the same number of vertices as its Minkowski sum with $Q$.

The total distance (or Wiener index) of a connected graph $G$ is the sum of all distances between unordered pairs of vertices of $G$ . DeLaViña and Waller [‘Spanning trees with many leaves and average distance’, Electron. J. Combin.15(1) (2008), R33, 14 pp.] conjectured in 2008 that if $G$ has diameter $D>2$ and order $2D+1$ , then the total distance of $G$ is at most the total distance of the cycle of the same order. In this note, we prove that this conjecture is true for 2-connected graphs.

Twenty one samples of relatively pure tubular halloysites (HNTs) from localities in Australia, China, New Zealand, Scotland, Turkey and the USA have been investigated by X-ray diffraction (XRD), infrared spectroscopy (IR) and electron microscopy. The halloysites occur in cylindrical tubular forms with circular or elliptical cross sections and curved layers and also as prismatic tubular forms with polygonal cross sections and flat faces. Measurements of particle size indicate a range from 40 to 12,700 nm for tube lengths and from 20 to 600 nm for diameters. Size distributions are positively skewed with mean lengths ranging from 170 to 950 nm and mean diameters from 50 to 160 nm. Cylindrical tubes are systematically smaller than prismatic ones. Features related to order/ disorder in XRD patterns e.g. as measured by a ‘cylindrical/prismatic’ (CP) index and IR spectra as measured by an ‘OH-stretching band ratio’ are related to the proportions of cylindrical vs. prismatic tubes and correlated with other physical measurements such as specific surface area and cation exchange capacity. The relationships of size to geometric form, along with evidence for the existence of the prismatic form in the hydrated state and the same 2M 1 stacking sequence irrespective of hydration state (i.e. 10 vs. 7 Å) or form, suggests that prismatic halloysites are the result of continued growth of cylindrical forms.

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 $H$ be a group. The co-maximal graph of subgroups of $H$ , denoted by $\Gamma \left( H \right)$ , is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma \left( H \right)$ if and only if $H\,=\,LK$ . In this paper, we study the connectivity, diameter, clique number, and vertex chromatic number of $\Gamma \left( H \right)$ . For instance, we show that if $\Gamma \left( H \right)$ has no isolated vertex, then $\Gamma \left( H \right)$ is connected with diameter at most 3. Also, we characterize all finitely groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma \left( H \right)$ is connected, and moreover, the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.

The unitary Cayley graph of a ring $R$ , denoted $\Gamma \left( R \right)$ , is the simple graph defined on all elements of $R$ , and where two vertices $x$ and $y$ are adjacent if and only if $x\,-\,y$ is a unit in $R$ . The largest distance between all pairs of vertices of a graph $G$ is called the diameter of $G$ and is denoted by $\text{diam}\left( G \right)$ . It is proved that for each integer $n\,\ge \,1$ , there exists a ring $R$ such that $\text{diam}\left( \Gamma \left( R \right) \right)=n$ . We also show that $\text{diam}\left( \Gamma \left( R \right) \right)\in \left\{ 1,2,3,\infty\right\}$ for a ring $R$ with ${R}/{J\left( R \right)}\;$ self-injective and classify all those rings with $\text{diam}\left( \Gamma \left( R \right) \right)\,=\,1,\,2,\,3$ , and $\infty$ , respectively.

In this paper we study random Apollonian networks (RANs) and evolving Apollonian networks (EANs), in d dimensions for any d≥2, i.e. dynamically evolving random d-dimensional simplices, looked at as graphs inside an initial d-dimensional simplex. We determine the limiting degree distribution in RANs and show that it follows a power-law tail with exponent τ=(2d-1)/(d-1). We further show that the degree distribution in EANs converges to the same degree distribution if the simplex-occupation parameter in the nth step of the dynamics tends to 0 but is not summable in n. This result gives a rigorous proof for the conjecture of Zhang et al. (2006) that EANs tend to exhibit similar behaviour as RANs once the occupation parameter tends to 0. We also determine the asymptotic behaviour of the shortest paths in RANs and EANs for any d≥2. For RANs we show that the shortest path between two vertices chosen u.a.r. (typical distance), the flooding time of a vertex chosen uniformly at random, and the diameter of the graph after n steps all scale as a constant multiplied by log n. We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar central limit theorem for typical distances in EANs.