let g be a simple connected graph. the generalized polarity wiener index ofg is defined as the number of unordered pairs of vertices of g whosedistance is k. some formulas are obtained for computing the generalizedpolarity wiener index of the cartesian product and the tensor product ofgraphs in this article.

A k?coloring of an oriented graph G = (V; A) is an assignment c of one of the colors 1; 2; : : :; k to each vertex of the graph such that, for every arc (x; y) of G, c(x) 6 = c(y). The k?coloring is good if for every arc (x; y) of G there is no arc (z; t) 2 A such that c(x) = c(t) and c(y) = c(z). A k?coloring is said to be semi?strong if for every vertex x of G, c(z) 6 = c(t) for any pair fz; ...

in this paper, the weighted szeged indices of cartesian product and corona product of twoconnected graphs are obtained. using the results obtained here, the weighted szeged indices ofthe hypercube of dimension n, hamming graph, c4 nanotubes, nanotorus, grid, t− fold bristled,sunlet, fan, wheel, bottleneck graphs and some classes of bridge graphs are computed.

We prove that among finite graph algebras and among finite flat graph algebras, dualizability, full dualizability, strong dualizability and entropicity are all equivalent. Any finite (flat) graph algebra which is not dualizable must be inherently non-κ-dualizable for every infinite cardinal κ. A new, general method for proving strong duality is presented.

We prove that the product replacement graph on generating k-tuples of a simple group contains a large connected component if k ≥ 3. This is related to the recent conjecture of Diaconis and Graham. As an application, we also prove that the output of the product replacement algorithm (see [CLMNO]) in this case does not have a strong bias.

the corona product $gcirc h$ of two graphs $g$ and $h$ isobtained by taking one copy of $g$ and $|v(g)|$ copies of $h$;and by joining each vertex of the $i$-th copy of $h$ to the$i$-th vertex of $g$, where $1 leq i leq |v(g)|$. in thispaper, exact formulas for the eccentric distance sum and the edgerevised szeged indices of the corona product of graphs arepresented. we also study the conditions...

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V (G) into disjoint sets V1 ∪ . . . ∪ Vr, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by ...

In many real world problems, data sometimes comes from n agents (n≥2), multipolar information exists. For issues that are associated with uncertainty, this information can not be showed with the values of crisp, fuzzy, intuitionistic or bipolar. Graph is one of the mathematical models widely used in different sciences. Ambiguity in a graph where data depends on the n parameter can not be showed...

The sigma coindex is defined as the sum of squares differences between degrees all nonadjacent vertex pairs. In this paper, we propose some mathematical properties coindex. Later, present precise results for coindices various graph operations such tensor product, Cartesian lexicographic disjunction, strong union, join, and corona product.