Mathematics educators suggest that students of all ages need to participate in productive forms of mathematical argument (NCTM, 2000). Accordingly, we developed two complementary frameworks for analyzing the emergence of mathematical argumentation in one second-grade classroom. Children attempted to resolve contesting claims about the “space covered” by three different-looking rectangles of equ...

Let G be a self-complementary graph (w.) and ir its degree sbquence. Th ;n G has a 2-factor if and only if R 2 is graphic. This is achieved by ohtaming a structure theorem regarding S.C. graphs without a Z-factor. Another irrreresting corollary of the structure theorem is that if G is a s c. graph of order p 3 N with minimum degree at least p/4, then G has a ?-factor and the result is the best ...

the annihilating-ideal graph of a commutative ring $r$ is denoted by $ag(r)$, whose vertices are all nonzero ideals of $r$ with nonzero annihilators and two distinct vertices $i$ and $j$ are adjacent if and only if $ij=0$. in this article, we completely characterize rings $r$ when $gr(ag(r))neq 3$.

Visitors to museums and other cultural heritage sites encounter a wealth of exhibits in a variety of subject areas, but can explore only a small number of them. Moreover, there typically exists rich complementary information that can be delivered to the visitor about exhibits of interest, but only a fraction of this information can be consumed during the limited time of the visit. Recommender s...

a connected graph g is said to be neighbourly irregular graph if no two adjacent vertices of g have same degree. in this paper we obtain neighbourly irregular derived graphs such as semitotal-point graph, k^{tℎ} semitotal-point graph, semitotal-line graph, paraline graph, quasi-total graph and quasivertex-total graph and also neighbourly irregular of some graph products.

In this paper, we aim to design sparse D-optimal (determinantoptimal) pose-graph SLAM problems through the synthesis of sparse graphs with the maximum weighted number of spanning trees. Characterizing graphs with the maximum number of spanning trees is an open problem in general. To tackle this problem, several new theoretical results are established in this paper, including the monotone log-su...

it is necessary to generate the automorphism group of a chemical graph in computer-aidedstructure elucidation. an euclidean graph associated with a molecule is defined by a weightedgraph with adjacency matrix m = [dij], where for i≠j, dij is the euclidean distance between thenuclei i and j. in this matrix dii can be taken as zero if all the nuclei are equivalent. otherwise,one may introduce dif...

The reciprocal complementary Wiener number of a connected graph G is defined as ( ) { } ( ) ( ) | ∑ u v V G RCW G d d u v G ⊆ = + − , 1 1 , where ( ) V G is the vertex set. ( ) | d u v G , is the distance between vertices u and v, and d is the diameter of G. A tree is known as a caterpillar if the removal of all pendant vertices makes it as a path. Otherwise, it is called a non-caterpillar. Amo...

in this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. also, density and balanced irregular vague graphs are discussed and some of their properties are established. finally we give an application of vague digraphs.

Let G(V;E) be a graph. The common neighborhood graph (congraph) of G is a graph with vertex set V , in which two vertices are adjacent if and only if they have a common neighbor in G. In this paper, we obtain characteristics of congraphs under graph operations; Graph :::::union:::::, Graph cartesian product, Graph tensor product, and Graph join, and relations between Cayley graphs and its c...