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The EDGE-tenacity Te(G) of a graph G was de ned as Te(G) = min F E(G) f j F j + (G 􀀀 F)! (G 􀀀 F) g where the minimum is taken over all EDGE cutset F of G. We de ne G-F to be the graph induced by the EDGEs of E(G)􀀀 F,  (G􀀀 F) is the number of EDGEs in the largest component of the graph induced by G-F and! (G 􀀀 F) is the number of components of G􀀀 F. A set F  E(G) is said to be a Te-set of G if Te(G) = jFj+ (G􀀀 F)! (G􀀀 F) Each component has at least one EDGE. In this paper we introduce a new invariant EDGE-tenacity, for graphs. it is another vulnerability measure. we present several properties and bounds on the EDGE-tenacity. we also compute the EDGE-tenacity of some classes of graphs.

In this paper, we study both concepts of geodetic dominating and EDGE geodetic dominating sets and derive some tight upper bounds on the EDGE geodetic and the EDGE geodetic domination numbers. We also obtain attainable upper bounds on the maximum number of elements in a partition of a vertex set of a connected graph into geodetic sets, EDGE geodetic sets, geodetic dominating sets and EDGE geodetic dominating sets, respectively.

For a coloring c of a graph G, the EDGE-di erence coloring sum and EDGE-sum coloring sum with respect to the coloring c are respectively Σ c D(G) = Σ jc(a) 􀀀 c(b)j and Σ s S(G) = Σ (c(a) + c(b)), where the summations are taken over all EDGEs ab 2 E(G). The EDGE-di erence chromatic sum, denoted by Σ D(G), and the EDGE-sum chromatic sum, denoted by Σ S(G), are respectively the minimum possible values of Σ c D(G) and Σ c S(G), where the minimums are taken over all proper coloring of c. In this work, we study the EDGE-di erence chromatic sum and the EDGE-sum chromatic sum of graphs. In this regard, we present some necessary conditions for the existence of homomorphism between two graphs. Moreover, some upper and lower bounds for these parameters in terms of the fractional chromatic number are introduced as well.

‎A $(k,6)$-fullerene graph is a planar $3$-connected cubic graph whose faces are $k$-gons and hexagons‎. ‎The aim of this paper is to‎ study the EDGE metric dimension of $(3,6)$‎- ‎and $(4,6)$-fullerene graphs‎.

In this paper, we introduced the concept of EDGE geodetic sequences in graph and its generating function. Some general properties satisfied by this concept are studied. It is shown that for every generating function$$ G(x)=\sum_{i=1}^{\infty} {a}^{i-1}{x^{i-1}} \quad a\in N-\left\lbrace 1\right\rbrace,$$there exists a recurrence graph $G$ with EDGE geodetic decomposition $\pi=\{G_{1},G_{2},\ldots ,G_{n}\ldots\}$.