Given a graph G, the burning number of G is smallest integer k for which there are vertices $$x_1, x_2,\ldots ,x_k$$ such that $$(x_1,x_2,\ldots ,x_k)$$ sequence G. It has been shown problem NP-complete, even trees with maximum degree three, or linear forests. A t-unicyclic unicycle in unique vertex greater than two $$ t + 2 . In this paper, we first present bounds graphs, and then use numbers ...

A connected graph G = (V, E) is called a quasi-tree graph if there exists a vertex u0 ∈ V (G) such that G−u0 is a tree. A connected graph G = (V, E) is called a quasi-unicyclic graph if there exists a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph. Set T (n, k) := {G : G is a n-vertex quasi-tree graph with k pendant vertices}, and T (n, d0, k) := {G : G ∈ T (n, k) and there is a vertex ...

A connected graph G = (V, E) is called a quasi-tree graph if there exists a vertex u0 ∈ V (G) such that G−u0 is a tree. A connected graph G = (V, E) is called a quasi-unicyclic graph if there exists a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph. Set T (n, k) := {G : G is a n-vertex quasi-tree graph with k pendant vertices}, and T (n, d0, k) := {G : G ∈ T (n, k) and there is a vertex ...

For a graph $G = (V, E)$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_k}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j in pi$, whenever $i neq j$. The textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. We study the properties of upper domatic number and propose an up...

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph G⊕K2 to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.

We study the influence of edge subdivision on the convex domination number. We show that in general an edge subdivision can arbitrarily increase and arbitrarily decrease the convex domination number. We also find some bounds for unicyclic graphs and we investigate graphs G for which the convex domination number changes after subdivision of any edge in G.

We consider the number of vertex independent sets i(G). In general, the problem of determining the value of i(G) is NP -complete. We present several upper and lower bounds for i(G) in terms of order, size or independence number. We obtain improved bounds for i(G) on restricted graph classes such as the bipartite graphs, unicyclic graphs, regular graphs and claw-free graphs.

let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...