A conjecture of Fan and Raspaud [3] asserts that every bridgeless cubic graph contains three perfect matchings with empty intersection. Kaiser and Raspaud [6] suggested a possible approach to this problem based on the concept of a balanced join in an embedded graph. We give here some new results concerning this conjecture and prove that a minimum counterexample must have at least 32 vertices.

In this paper we determine when the join of two graphs is locally homogeneous. It is shown that the Cartesian product of a locally H1 graph and a locally H2 graph is locally (H1∪ H2). All graphs H of order at most 4 for which there are no locally H graphs are determined.

In11, Kulli and Janakiram initiate the concept of maximal domination in graphs. In this paper, we obtained some bounds characterizations. Also, estimate value number graph products such as join graphs, corona product, cartesian product strong product.

In this paper, we introduce the dependence graph of a lattice defined on the set of its join-irreducible elements. This graph, issued from the dependence relation on a lattice, is a nice structure encoding together the minimal generators and the canonical direct basis of a lattice. Then, we propose a new generation algorithm.

Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph EG as follows: Take G\L(G) as vertices of EG and join two distinct vertices x and y whenever [x,k y] 6= 1 and [y,k x] 6= 1 for all positive integers k. We call EG, the Engel graph of G. In this paper we study the graph theoretical properties of EG.

A graph is well-covered if all its maximal independent sets are of the same size (M. D. Plummer, 1970). A well-covered graph (with at least two vertices) is 1-wellcovered if the deletion of every vertex leaves a graph which is well-covered as well (J. W. Staples, 1975). In this paper, we provide new characterizations of 1-well-covered graphs, which we further use to build 1-well-covered graphs ...

A k-coloring of a graph G = (V,E) is a function c : V → C, where |C| = k. (Most often we use C = [k].) Vertices of the same color form a color class. A coloring is proper if adjacent vertices have different colors. A graph is k-colorable if there is a proper k-coloring. The chromatic number χ(G) of a graph G is the minimum k such that G is k-colorable. Let H and G be graphs. The disjoint union ...

This paper gives tight upper bound on the largest eigenvalue q (G) of the signless Laplacian of graphs with no paths of given order. Thus, let Sn,k be the join of a complete graph of order k and an independent set of order n− k, and let S n,k be the graph obtained by adding an edge to Sn,k. The main result of the paper is the following theorem: Let k ≥ 1, n ≥ 7k, and let G be a graph of order n...

The crossing number of a graph G, cr(G), is defined as the smallest possible edge-crossings in drawing G plane. There are almost no results concerning join disconnected 6-vertex with cycle. main aim this paper to give product Q+Cn for Q consisting two 3-cycles, where Cn cycle on n vertices.

We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetković and Rowlinson from 1...