For a simple graph G, a vertex labeling φ : V (G) → {1, 2, · · · , k} is called k-labeling. The weight of an edge xy in G, denoted by wπ(xy), is the sum of the labels of end vertices x and y, i.e. wφ(xy) = φ(x) + φ(y). A vertex k-labeling is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f , there is wφ(e) 6= wφ(f). The minimum k for which the g...

An edge magic total labeling of a graph G(V,E) with p vertices and q edges is a bijection f from the set of vertices and edges to such that for every edge uv in E, f(u) + f(uv) + f(v) is a constant k. If there exist two constants k1 and k2 such that the above sum is either k1 or k2, it is said to be an edge bimagic total labeling. A total edge magic (edge bimagic) graph is called a super edge m...

let $r$ be a commutative ring with zero-divisor set $z(r)$. the total graph of $r$, denoted by$t(gamma(r))$, is the simple (undirected) graph with vertex set  $r$ where two distinct vertices areadjacent if their sum lies in $z(r)$.this work considers minimum zero-sum $k$-flows for $t(gamma(r))$.both for $vert rvert$ even and the case when $vert rvert$ is odd and $z(g)$ is an ideal of $r$it is s...

An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring of G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ a (G). We prove that χ a (G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges w...

A matched sum graph G1MG2 of two graphs G1 and G2 of the same order n is obtained by adding to the union (or sum) of G1 and G2 a set M of n independent edges which join vertices in V (G1) to vertices in V (G2). When G1 and G2 are isomorphic, G1MG2 is just a permutation graph. In this work we derive bounds for the k-restricted edge connectivity λ(k) of matched sum graphs G1MG2 for 2 ≤ k ≤ 5, and...

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all edges incident $x$. The chromatic number $G$, denoted by $\chi_{la}(G)$, minimum distinct labels labelings $G$. In this...

A graph G is k–weighted–list–antimagic if for any vertex weighting ω : V (G) → R and any list assignment L : E(G)→ 2R with |L(e)| ≥ |E(G)|+k there exists an edge labeling f such that f(e) ∈ L(e) for all e ∈ E(G), labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this pa...

If F is a (possibly infinite) subset of an abelian group Γ, then we define f(F,Γ) as the smallest natural number such that every f(F,Γ)edge-connected (finite) graph G has a flow where all flow values are elements in F . We prove that f(F,Γ) exists if and only if some odd sum of elements in F equals some even sum. We discuss various instances of this problem. We prove that every 6-edge-connected...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

We analyze the simple greedy algorithm that iteratively removes the endpoints of a maximum-degree edge in a graph, where the degree of an edge is the sum of the degrees of its endpoints. This algorithm provides a 2-approximation to the minimum edge dominating set and minimum maximal matching problems. We refine its analysis and give an expression of the approximation ratio that is strictly less...