A path in an edge-colored graph is rainbow if no two edges of it are colored the same. The graph is said to be rainbow connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph is strong rainbow connected. We consider the complexity of the problem of deciding if a given edge-colored graph is rainbow or stro...

The k-th semi total point graph of a graph G, , ‎is a graph‎ obtained from G by adding k vertices corresponding to each edge and‎ connecting them to the endpoints of edge considered‎. ‎In this paper‎, a formula for Laplacian polynomial of in terms of‎ characteristic and Laplacian polynomials of G is computed‎, ‎where is a connected regular graph‎.The Kirchhoff index of is also computed‎.

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...

An edge geodetic set of a connected graph G of order p ≥ 2 is a set S ⊆ V (G) such that every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum cardinality of its edge geodetic sets and any edge geodetic set of cardinality g1(G) is a minimum edge geodetic set of G or an edge geodetic basis of G. An edge geodetic set S in ...

In a graph, by definition, the weight of a (proper) coloring with positive integers is the sum of the colors. The chromatic sum is the minimum weight, taken over all the proper colorings. The minimum number of colors in a coloring of minimum weight is the cost chromatic number or strength of the graph. We derive general upper bounds for the strength, in terms of a new parameter of representatio...

Effective tools are crucial for visualizing large quantities of information. While developing these tools, numerous graph drawing problems emerge. We present solutions for reducing clutter in a radial visualization of a bipartite graph representing the alerts generated by an IDS protecting a computer network. Our solutions rely essentially on (i) unambiguous edge bundling to reduce the number o...

Given an integer r > 0, let G r = (V; E) denote a graph consisting of a simple nite undirected connected nontrivial graph G together with r isolated vertices K r. Let L : V ! Z + denote a labelling of the vertices of G r with distinct positive integers. Then G r is said to be a sum graph if there exists a labelling L such that for every distinct vertex pair u and v of V , (u; v) 2 E if and only...

In a graph G, we say a cycle C : v1, v2, . . . , vk, v1 is chorded if its vertices induce an additional edge (chord) which is not an edge of the cycle. The cycle C is doubly chorded if there are at least two such chords. In this paper we show a sharp degree sum condition that implies the existence of k vertex disjoint doubly chorded cycles in a graph. © 2015 Elsevier B.V. All rights reserved.

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

In this paper, we propose a new edge-based matching kernel for graphs by using discrete-time quantum walks. To this end, we commence by transforming a graph into a directed line graph. The reasons of using the line graph structure are twofold. First, for a graph, its directed line graph is a dual representation and each vertex of the line graph represents a corresponding edge in the original gr...