In 1980, Enomoto et al. proposed the conjecture that every tree is a super (a, 0)-edge-antimagic total graph. In this paper, we give a partial sup- port for the correctness of this conjecture by formulating some super (a, d)- edge-antimagic total labelings on a subclass of subdivided stars denoted by T(n, n + 1, 2n + 1, 4n + 2, n5, n6, . . . , nr) for different values of the edge- antimagic labeling parameter d, where n ≥ 3 is odd, nm = 2m−4(4n+1)+1, r ≥ 5 and 5 ≤ m ≤ r.

Enomoto, Llado, Nakamigawa and Ringel (1998) defined the concept of a super (a, 0)-edge-antimagic total labeling and proposed the conjecture that every tree is a super (a, 0)-edge-antimagic total graph. In the support of this conjecture, the present paper deals with different results on super (a, d)-edge-antimagic total labeling of subdivided stars for d ∈ {0, 1, 2, 3}.

A graph $G$ with $p$ vertices and $q$ edges, vertex set $V\left(G\right)$ and edge set $E\left(G\right)$, is said to be super vertex-graceful (in short SVG), if there exists a function pair $(f,f^{+})$ where $f$ is a bijection from $V\left(G\right)$ onto $P$, $f^{+}$ is a bijection from $E\left(G\right)$ onto $Q$, $f^{+}\left((u,v)\right)=f\left(u\right)+f\left(v\right)$ for any $(u,v)\in E\left(G\right)$, $$Q=\left\{\begin{array}{cc}\{\pm 1,\cdots ,\pm \frac{1}{2}q\},\hfill & \text{if}q\text{is}\text{even,}\hfill \\ \{0,\pm 1,\cdots ,\pm \frac{1}{2}(q-1)\},\hfill & \text{if}q\text{is}\text{odd,}\hfill \end{array}\right.$$ and $$P=\left\{\begin{array}{cc}\{\pm 1,\cdots ,\pm \frac{1}{2}p\},\hfill & \text{if}p\text{is}\text{even,}\hfill \\ \{0,\pm 1,\cdots ,\pm \frac{1}{2}(p-1)\},\hfill & \text{if}p\text{is}\text{odd.}\hfill \end{array}\right.$$ We determine here families of unicyclic graphs that are super vertex-graceful.

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. Some constructions of supermagic labellings of regular graphs are described. Supermagic regular complete multipartite graphs and supermagic cubes are characterized.

Let γ(G) and ${\gamma }_{2,2}\left(G\right)$ denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that $\left(2(\gamma \left(T\right)+1)\right)/3\le {\gamma }_{2,2}\left(T\right)\le 2\gamma \left(T\right)$. Moreover, we characterize all the trees achieving the equalities.

In this note we prove that F (2, 2, 4) = 13.

In this paper we show bounds for the adjacent eccentric distance sum of graphs in terms of Wiener index, maximum degree and minimum degree. We extend some earlier results of Hua and Yu [Bounds for the Adjacent Eccentric Distance Sum, International Mathematical Forum. Vol. 7 (2O02) no. 26. 1280-1294]. The adjaceni eccentric distance sum index of the graph G is defined as [...] where ε(υ) is the eccentricity of the vertex υ, deg(υ) is the degree of the vertex υ and D(υ) = ∑u∊v(G) d (u,υ)is the sum...

The investigation of the counting function of the set of integral elements, in an algebraic number field, with factorizations of at most k different lengths gives rise to a combinatorial constant depending only on the class group of the number field and the integer k. In this paper the value of these constants, in case the class group is an elementary p-group, is estimated, and determined under additional conditions. In particular, it is proved that for elementary 2-groups these constants are equivalent...