the wiener index is a graph invariant that has found extensive application in chemistry. inaddition to that a generating function, which was called the wiener polynomial, who’sderivate is a q-analog of the wiener index was defined. in an article, sagan, yeh and zhang in[the wiener polynomial of a graph, int. j. quantun chem., 60 (1996), 959969] attainedwhat graph operations do to the wiener po...

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a bcoloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G H, the lexicographic product G[H] and the direct produ...

The strong chromatic number, χS(G), of an n-vertex graph G is the smallest number k such that after adding kdn/ke−n isolated vertices to G and considering any partition of the vertices of the resulting graph into disjoint subsets V1, . . . , Vdn/ke of size k each, one can find a proper k-vertex-coloring of the graph such that each part Vi, i = 1, . . . , dn/ke, contains exactly one vertex of ea...

Let G and H be connected graphs. The tensor product G + H is a graph with vertex set V(G+H) = V (G) X V(H) and edge set E(G + H) ={(a , b)(x , y)| ax ∈ E(G) & by ∈ E(H)}. The graph H is called the strongly triangular if for every vertex u and v there exists a vertex w adjacent to both of them. In this article the tensor product of G + H under some distancebased topological indices are investiga...

It was recently proved that every planar graph is a subgraph of the strongproduct path and with bounded treewidth. This paper surveys generalisationsof this result for graphs on surfaces, minor-closed classes, various nonminor-closed classes polynomial growth. We then explorehow product structure might be applicable to more broadly defined graphclasses. In particular, we characterise when class...

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 narumi-katayama index was the first topological index defined‎ ‎by the product of some graph theoretical quantities‎. ‎let $g$ be a ‎simple graph with vertex set $v = {v_1,ldots‎, ‎v_n }$ and $d(v)$ be‎ ‎the degree of vertex $v$ in the graph $g$‎. ‎the narumi-katayama ‎index is defined as $nk(g) = prod_{vin v}d(v)$‎. ‎in this paper,‎ ‎the narumi-katayama index is generalized using a $n$-ve...

Let $A$ be a commutative ring with nonzero identity, and $1leq n

A semideenite relaxation (?) for the problem of nding the maximum number (?) of edges in a complete bipartite subgraph of a bipartite graph ? = (V1 V2; E) is considered. For a large class of graphs, the relaxation is better than the LP-relaxation described in 8]. It is shown that (?) is bounded from above by the Lovv asz theta function (LQ(?)) of the graph LQ(?) related to the line graph of ?. ...

A semidefinite relaxation σ(Γ) for the problem of finding the maximum number κ(Γ) of edges in a complete bipartite subgraph of a bipartite graph Γ = (V1 ∪ V2, E) is considered. For a large class of graphs, the relaxation is better than the LP-relaxation described in [9]. It is shown that σ(Γ) is bounded from above by the Lovász theta function θ(LQ(Γ)) of the graph LQ(Γ) related to the line grap...