For a connected graph G of order n, a subset S of vertices of G is a monophonic set of G if each vertex v in G lies on a x-y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A monophonic set of cardinality m(G) is called a m–set of G. A set S of vertices of a connected graph G is an open m...

let $g$ be a simple graph of order $n$ and size $m$.the edge covering of $g$ is a set of edges such that every vertex of $g$ is incident to at least one edge of the set. the edge cover polynomial of $g$ is the polynomial$e(g,x)=sum_{i=rho(g)}^{m} e(g,i) x^{i}$,where $e(g,i)$ is the number of edge coverings of $g$ of size $i$, and$rho(g)$ is the edge covering number of $g$. in this paper we stud...

We show the hardness of the geodetic hull number for chordal graphs.

Let G be a subgraph of Kn. The graph obtained from G by replacing each edge with a 3-cycle whose third vertex is distinct from other vertices in the configuration is called a T (G)-triple. An edge-disjoint decomposition of 3Kn into copies of T (G) is called a T (G)-triple system of order n. If, in each copy of T (G) in a T (G)triple system, one edge is taken from each 3-cycle (chosen so that th...

This paper is dedicated to propose an algorithm in order to generate the certain isomers of some well-known fullerene bases. Furthermore, we enlist the bipartite edge frustration correlated with some of symmetrically distinct innite families of fullerenes generated by the oered process.

Given graphs G and H and a positive number b, a weighted (H, b)-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms an H-subgraph. We assign a weight of b to each H-subgraph in the decomposition and a weight of 1 to single edges. The total weight of the decomposition is the sum of the weights of all elements in the decomposition. Let φ(n,H...

Let G be a graph whose edges are colored with k colors, and H=(H1,⋯,Hk) be a k-tuple of graphs. A monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic copy of Hi in color i, for some 1≤i≤k. Let φk(n,H) be the smallest number ϕ, such that, for every order-n graph and every k-edge-coloring, there is a monochrom...

Given a connected graph and a vertex x ∈ V (G), the geodesic graph Px(G) has the same vertex set as G with edges uv iff either v is on an x − u geodesic path or u is on an x − v geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of Km,n with itself, where m,n ≥ 4, is e...