Many authors studied the graph theory in connection with commutative semigroups and commutative and noncommutative rings as we can refer to references. For example, Beck 1 associated to any commutative ring R its zero-divisor graph G R whose vertices are the zero-divisors of R including 0 , with two vertices a, b joined by an edge in case ab 0. Also, DeMeyer et al. 2 defined the zero-divisor gr...

Let u and v be any two distinct nodes of an undirected graph G, which is k-connected. For 1 ≤ w ≤ k, a w-container C(u, v) of a k-connected graph G is a set of w-disjoint paths joining u and v. A w-container C(u, v) of G is a w-container if it contains all the nodes of G. A graph G is w-connected if there exists a w-container between any two distinct nodes. A bipartite graph G is w-laceable if ...

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colors needed to color its edges, so that every pair of vertices is connected by at least one path in which no two edges are colored the same. Our main result is that rc(G) ≤ dn2 e for any 2-connected graph with at least three vertices. We conjecture that rc(G) ≤ n/κ + C for a κ-connected graph G of order n, w...

For a connected graph G of diameter d and an integer k with 1 ≤ k ≤ d, a radio k-coloring of G is an assignment c of colors (positive integers) to the vertices of G such that d(u, v) + |c(u)− c(v)| ≥ 1 + k for every two distinct vertices u and v of G, where d(u, v) is the distance between u and v. The value rck(c) of a radio k-coloring c of G is the maximum color assigned to a vertex of G. The ...

For positive integers d and m, let Pd,m(G) denote the property that between each pair of vertices of the graph G , there are m internally vertex disjoint paths of length at most d. For a positive integer t a graph G satisfies the minimum generalized degree condition δt(G) ≥ s if the cardinality of the union of the neighborhoods of each set of t vertices of G is at least s. Generalized degree co...

Denote by σ̄k = min {d(x1)+d(x2)+ · · ·+d(xk)−|N(x1)∩N(x2)∩· · ·∩N(xk)| | x1, x2, · · · , xk are k independent vertices in G}. Let n and m denote the number of vertices and edges of G. For any connected graph G, we give a polynomial algorithm in O(nm) time to either find two disjoint paths P1 and P2 such that |P1|+ |P2| ≥ min{σ̄4, n} or output G = ∪i=1Gi such that for any i, j ∈ {1, 2, ..., k} (k...

A connected graph is double connected if its complement is also connected The following Ramsey type theorem is proved in this paper There exists a function h n de ned on the set of integers exceeding three such that every double connected graph on at least h n vertices must contain as an induced subgraph a special double connected graph which is either one of the following graphs or the complem...

Given an ordered partition Π = {P1, P2, ..., Pt} of the vertex set V of a connected graph G = (V,E), the partition representation of a vertex v ∈ V with respect to the partition Π is the vector r(v|Π) = (d(v, P1), d(v, P2), ..., d(v, Pt)), where d(v, Pi) represents the distance between the vertex v and the set Pi. A partition Π of V is a resolving partition of G if different vertices of G have ...

If u and v are vertices of a graph, then d(u, v) denotes the distance from u to v. Let S = {v1, v2, . . . , vk} be a set of vertices in a connected graph G. For each v ∈ V (G), the k-vector cS(v) is defined by cS(v) = (d(v, v1), d(v, v2), · · · , d(v, vk)). A dominating set S = {v1, v2, . . . , vk} in a connected graph G is a metric-locatingdominating set, or an MLD-set, if the k-vectors cS(v) ...

‎the second multiplicative zagreb coindex of a simple graph $g$ is‎ ‎defined as‎: ‎$${overline{pi{}}}_2left(gright)=prod_{uvnotin{}e(g)}d_gleft(uright)d_gleft(vright),$$‎ ‎where $d_gleft(uright)$ denotes the degree of the vertex $u$ of‎ ‎$g$‎. ‎in this paper‎, ‎we compare $overline{{pi}}_2$-index with‎ ‎some well-known graph invariants such as the wiener index‎, ‎schultz‎ ‎index‎, ‎eccentric co...