We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in AT-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in perm...

A vertex irregular total k-labeling of a graph G with vertex set V and edge set E is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of G, denoted by tvs(G)is the minimum value of the largest label k over all such irregular assignment. In this paper, we study the to...

the prime graph $gamma(g)$ of a group $g$ is a graph with vertex set $pi(g)$, the set of primes dividing the order of $g$, and two distinct vertices $p$ and $q$ are adjacent by an edge written $psim q$ if there is an element in $g$ of order $pq$. let $pi(g)={p_{1},p_{2},...,p_{k}}$. for $pinpi(g)$, set $deg(p):=|{q inpi(g)| psim q}|$, which is called the degree of $p$. we also set $d(g):...

the non-commuting graph $nabla(g)$ of a non-abelian group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we 'll prove that if $g$ is a finite group with $nabla(g)congnabla(bs_{n})$, then $g cong bs_{n}$, where $bs_{n}$ is the symmetric group of degre...

Based on the differential transformation method, a semi-analytical solution is developed for free vibration and modal stress analyses of variable thickness two-directional functionally graded circular plates with restrained edges, resting on elastic foundations. Variations of the material and geometry parameters are monitored by five distinct exponential functions. The presented non-dimensional...

Let R be a commutative ring with identity such that R admits at least two maximal ideals. In this article, we associate a graph with R whose vertex set is the set of all proper ideals I of R such that I is not contained in the Jacobson radical of R and distinct vertices I and J are joined by an edge if and only if I and J are not comparable under the inclusion relation. The aim of this article ...

A decomposition of a graph G is a set D = {H1, · · · , Hk} of pairwise edge-disjoint subgraphs of G that cover the set of edges of G. If Hi is isomorphic to a fixed graph H, for 1 ≤ i ≤ k, then we say that D is an H-decomposition of G. In this work, we study the case where H is a path of fixed length. For that, we first decompose the given graph into trails, and then we use a disentangling lemm...

Definition. Let Kn denote the complete graph of n vertices. The complete multigraph λKn is the complete graph Kn in which every edge is taken λ times. The t-foil (or the t-windmill) is a graph of t edge-disjoint K3’s with a common vertex and the common vertex is called the center of the t-foil. In particular, the 2-foil, the 3-foil, and the 4-foil are called the bowtie, the trefoil, and the qua...

Thermal treatment of supersaturated Ti1!xAlxN films (x " 0.67) with a dominant ternary cubic-phase were performed in the 700– 1000 !C range. Grazing incidence X-ray diffraction (GIXRD) shows that, for annealing temperatures up to 800 !C, the film structure undergoes the formation of coherent cubic AlN (c-AlN) and TiN (c-TiN) nanocrystallites via spinodal decomposition and, at higher temperature...