Let $G= (V,E)$ be a $(p,q)$-graph. A bijection $f: Eto{1,2,3,ldots,q }$ is called an edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ where $f^+(u) = sum_{uwin E} f(uw)$. Moreover, a bijection $f: Eto{1,2,3,ldots,q }$ is called a semi-edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ or $f^+(u)=f^+(v)$. A graph that admits an  ...

In this paper, we consider 2.5D drawing of a pair of trees which are connected by some edges, representing relationships between nodes, as an attempt to develop a tool for analyzing pairwise hierarchical data. We consider two ways of drawing such a graph, called parallel and perpendicular drawings, where the graph appears as a bipartite graph viewed from two orthogonal angles X and Y . We defin...

The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) ...

A (p,q) graph G is said to admit n order triangular sum labeling if its vertices can be labeled by non negative integers such that the induced edge labels obtained by the sum of the labels of end vertices are the first q n order triangular numbers. A graph G which admits n order triangular sum labeling is called n order triangular sum graph. In this paper we prove that paths, combs, stars, subd...

let $r$ be a commutative ring with identity. an ideal $i$ of a ring $r$is called an annihilating ideal if there exists $rin rsetminus {0}$ such that $ir=(0)$ and an ideal $i$ of$r$ is called an essential ideal if $i$ has non-zero intersectionwith every other non-zero ideal of $r$. thesum-annihilating essential ideal graph of $r$, denoted by $mathcal{ae}_r$, isa graph whose vertex set is the set...

The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to vertex e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The main result of the pap...

Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),(1≤i≠j≤n). The maximum cardinality of π...

Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to (including itself) is positive. Then weights over at least $-\frac{n^2}{25}$. Also we provide an example weighted described properties $-(1+o(1))\frac{n^2}{8(1 + \sqrt{2})^2}$.
 The previous best known bounds were $-\frac{n^2}{16}$ $-(1+o(1))\frac{n^2}{54}$...

An injective function f : V (G)→ {0, 1, 2, . . . , q} is an odd sum labeling if the induced edge labeling f∗ defined by f∗(uv) = f(u) + f(v), for all uv ∈ E(G), is bijective and f∗(E(G)) = {1, 3, 5, . . . , 2q − 1}. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper, we have studied the odd sum property of the subdivision of the triangular snake, quadrilatera...

Let G be a directed graph with an unknown flow on each edge such that the following flow conservation constraint is maintained: except for sources and sinks, the sum of flows into a node equals the sum of flows going out of a node. Given a noisy measurement of the flow on each edge, the problem we address, which we call the MOST PROBABLE FLOW ESTIMATION problem (MPFE), is to estimate the most p...