Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

In the Colored Clustering problem, one is asked to cluster edge-colored (hyper-)graphs whose colors represent interaction types. More specifically, goal select as many edges possible without choosing two that share an endpoint and are colored differently. Equivalently, can also be described assigning vertices in a way fits edge-coloring well possible. As this problem NP-hard, we build on previo...

let $d$ be a digraph with vertex set $v(d)$. for vertex $vin v(d)$, the degree of $v$,denoted by $d(v)$, is defined as the minimum value if its out-degree and its in-degree.now let $d$ be a digraph with minimum degree $deltage 1$ and edge-connectivity$lambda$. if $alpha$ is real number, then the zeroth-order general randic index is definedby $sum_{xin v(d)}(d(x))^{alpha}$. a digraph is maximall...

Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms for computing the hyperbolicity number of a graph (the smaller, the more tree-like) have running time O(n), where n is the number of graph ...

In graph theory, as in many fields of mathematics, one is often interested in finding the maxima or minima of certain functions and identifying the points of optimality. We consider a variety of functions on graphs and hypegraphs and determine the structures that optimize them. A central problem in extremal (hyper)graph theory is that of finding the maximum number of edges in a (hyper)graph tha...

A hypergraph is a pair pV, Eq where V is a finite set and E Ď 2 is called the set of hyper-edges. An output-polynomial algorithm for C Ď 2 is an algorithm that lists without repetitions all the elements of C in time polynomial in the sum of the size of H and the accumulated size of all the elements in C. Whether there exists an output-polynomial algorithm to list all the inclusion-wise minimal ...

.............................................................................................. I Acknowledgements ............................................................................ II ركشو ءادهآ .............................................................................................. IV List of Figures .................................................................................