An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . ,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. A conjecture of Ringel (see [4]) states that every connected graph, but K2,...

We examine the influence of heterogeneous curing rates for a SIS model, used for malware spreading on the Internet, information dissemination in unreliable networks, and propagation of failures in networks. The topology structures considered are the regular graph which represents the homogenous network structures and the complete bi-partite graph which represents the hierarchical network struct...

A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a nite number of isolated vertices to it will always yield a sum graph, and the sum number (G) of G is the smallest number of isolated vertices that...

Let R be a ring with unity. The graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Let Γ2(R) is the subgraph of Γ(R) induced by the non-unit elements. H.R. Maimani et al. [H.R. Maimani et al., Comaximal graph of commutative rings, J. Algebra 319 (2008) 1801-1808] proved that: “If R is a commutative ring with unity a...

A Roman dominating function on a graph G = (V,E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman dominating function is the value f(G) = ∑ u∈V f(u). The Roman domination number of G is the minimum weight of a Roman dominating function on G. The Roman bondage number of a nonempty ...

We say that a digraph D is competitive if any pair of vertices has common out-neighbor in and graph G competitively orientable there exists orientation G. The competition defined as the with vertex set V(D) an edge uv only u v compete D. notion arose while studying whose graphs are complete. derive some useful properties show complete order n n≥7. Then we completely characterize multipartite te...

An n-player, finite, probabilistic game with perfect information can be presented as a 2n-partite graph. For Can’t Stop, the graph is cyclic and the challenge is to determine the game-theoretical values of the positions in the cycles. We have presented our success on tackling one-player Can’t Stop and two-player Can’t Stop. In this article we study the computational solution of multi-player Can...

A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n2/4 ...

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of hypergraphs $\mathcal{F}$, the Turán number ex$_r^{lin}(n,\mathcal{F})$ maximum on $n$ vertices that does not contain any member $\mathcal{F}$ as subgraph.
 Let $K_l$ be complete graph with $l$ and $r\geq 2$. The $r$-expansion $r$-graph $K_l^+$ obtained from by enlarging each edge $r-2...