For any graph G, let V (G) and E(G) denote the vertex set and the edge set of G respectively. The Boolean function graph B(G, L(G), NINC) of G is a graph with vertex set V (G) ∪ E(G) and two vertices in B(G, L(G), NINC) are adjacent if and only if they correspond to two adjacent vertices of G, two adjacent edges of G or to a vertex and an edge not incident to it in G. For brevity, this graph is denoted by B1(G). In this paper, we determine domination number, independent, connected, total, point-set, restrained, split and non-split domination numbers in the complement B1(G) of B1(G) and obtain bounds for the above numbers.
With an increase in growth irradiance (from 15 to 100 % of full sunlight, I15 to I100), the maximum net photosynthetic rate (Pmax), compensation (CI) and saturation irradiances of A. annua increased. At full sunlight, A. annua had a high capacity of photosynthesis, while at low irradiance it maintained a relatively high Pmax with a low CI. The height and diameter growth, total and leaf biomass, and artemisinin content of A. annua decreased with the decrease in irradiance, which might be connected with lower photosynthesis at lower than at higher irradiance. Irradiances changed biomass allocations of A. annua. The leaf/total mass ratio of A. annua increased with decreasing irradiance, but the root/total mass ratio and root/above-ground mass generally increased with increasing irradiance. Thus A. annua can grow in both weak and full sunlight. However, high yield of biomass and artemisinin require cultivation in an open habitat with adequate sunshine. and M. L. Wang ... [et al.].
The eccentricity e(v) of a vertex v is the distance from v to a vertex farthest from v, and u is an eccentric vertex for v if its distance from v is d(u, v) = e(v). A vertex of maximum eccentricity in a graph G is called peripheral, and the set of all such vertices is the peripherian, denoted PeriG). We use Cep(G) to denote the set of eccentric vertices of vertices in C(G). A graph G is called an S-graph if Cep(G) = Peri(G). In this paper we characterize S-graphs with diameters less or equal to four, give some constructions of • S-graphs and investigate S-graphs with one central vertex. We also correct and generalize some results of F. Gliviak.
Let G be a finite group. The intersection graph ΔG of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected., Xuanlong Ma., and Obsahuje seznam literatury
The reverse Wiener index of a connected graph $G$ is defined as \[ \Lambda (G)=\frac {1}{2}n(n-1)d-W(G), \] where $n$ is the number of vertices, $d$ is the diameter, and $W(G)$ is the Wiener index (the sum of distances between all unordered pairs of vertices) of $G$. We determine the $n$-vertex non-starlike trees with the first four largest reverse Wiener indices for $n\ge 8$, and the $n$-vertex non-starlike non-caterpillar trees with the first four largest reverse Wiener indices for $n\ge 10$.
In this paper we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks {\it 32} (1998), 199--206). A paired-dominating set of a graph $G$ with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of $G$, denoted by $\gamma _{{\rm pr}}(G)$, is the minimum cardinality of a paired-dominating set of $G$. The graph $G$ is paired-domination vertex critical if for every vertex $v$ of $G$ that is not adjacent to a vertex of degree one, $\gamma _{{\rm pr}}(G - v) < \gamma _{{\rm pr}}(G)$. We characterize the connected graphs with minimum degree one that are paired-domination vertex critical and we obtain sharp bounds on their maximum diameter. We provide an example which shows that the maximum diameter of a paired-domination vertex critical graph is at least $\frac 32(\gamma _{{\rm pr}}(G) - 2)$. For $\gamma _{{\rm pr}}(G) \le 8$, we show that this lower bound is precisely the maximum diameter of a paired-domination vertex critical graph.