A novel domination invariant defined by Boutrig and Chellali in the recent: total vertex-edge domination. In this paper we obtain an improved upper bound of total vertex edge-domination number of a tree. If is a connected tree with order , then ( ) ≤ 3 ⁄ with = 6⌈ 6 ⁄ ⌉ and we characterize the trees attaining this upper bound. Furthermore we provide a characterization of trees with ( ) = ( ).

In this paper, we computed the Tutte polynomials of (2,n)-torus knots and introduced general formulas for this process. Firstly, we obtained the isomorphic graphs and dual graphs of (2,n)-torus knots from their regular diagrams. Then, we computed the Tutte polynomials by these graphs. Finally, we obtained that the Tutte polynomials of the isomorphic graphs and dual graphs of (2,n)-torus knots are equivalent to each other. Moreover we computed the Tutte polynomials for signed graphs, which their edges are each labelled with a sign {+1 or − 1}, of (2,n)-torus knots. we obtained two generalizations for these graphs.

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An edge e ev-dominates a vertex v which is a vertex of e, as well as every vertex adjacent to v. A subset D ⊆ E is an edge-vertex dominating set (in simply, ev-dominating set) of G, if every vertex of a graph G is ev-dominated by at least one edge of D. The minimum cardinality of an evdominating set is named with ev-domination number and denoted by γev(G). A subset D ⊆ E is a total edge-vertex dominating set (in simply, total ev-dominating set) of G, if D is an ev-dominating set and every edge of D shares an endpoint with other edge of D. The total ev-domination number of a graph G is denoted with γ t ev (G) and it is equal to the minimum cardinality of a total ev-dominating set. In this paper, we initiate to study total edge-vertex domination. We first show that calculating the number γ t ev (G) for bipartite graphs is NP-hard. We also show the upper bound γ t ev (T ) ≤ (n − l + 2s − 1)/2 for the total ev-domination number of a tree T , where T has order n, l leaves and s support vertices and we characterize the trees achieving this upper bound. Finally, we obtain total ev-domination number of paths and cycles.Mathematics Subject Classification. 05C69.

Let T n be a twist knot with n half-twists and G n be the graph of T n . The closed neighborhood N[v] of a vertex v in G n , which included at least one colored vertex for each color in a proper n-coloring of G n , is called a rainbow neighborhood. There are different types of graph coloring in the literature. We consider some of these types in here. In this paper, we determine the chromatic number of graphs of twist knots and study rainbow neighborhood of graphs of twist knots. We determine the rainbow neighborhood number and the fading number of them. Furthermore, we determine coupon coloring and the coupon coloring number of graphs of twist knots.