We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let [Formula: see text] be unicyclic graph with unique odd cycle and its ideal. compute exact values all symbolic defects using describe one method to find quasi-polynomial associated ideal text]. classify class graphs when some power maximal annihilates any fixed Also those graphs, we Hilbert function module

The harmonic index of a graph G is defined as the sum of the weights 2 d.u/Cd.v/ of all edges uv of G, where d.u/ denotes the degree of a vertex u in G. In this paper, we present the minimum harmonic indices for unicyclic and bicyclic graphs with n vertices and matching number m (2 m bn2 c), respectively. The corresponding extremal graphs are also characterized. 2000 Mathematics Subject Classif...

Orthogonal representations are used to show that complements of certain sparse graphs have (positive semidefinite) minimum rank at most 4. This bound applies to the complement of a 2-tree and to the complement of a unicyclic graph. Hence for such graphs, the sum of the minimum rank of the graph and the minimum rank of its complement is at most two more than the order of the graph. The minimum r...

Topological indices are a class of numerical invariants that predict certain physical and chemical properties molecules. Recently, two novel topological indices, named as Sombor index reduced index, were introduced by Gutman, defined where denotes the degree vertex in . In this paper, our aim is to order trees, unicyclic graphs, bicyclic graphs tricyclic with respect index. We determine first f...

In an ordinary edge-coloring of a graph each color appears at each vertex at most once. An f -coloring is a generalized edge-coloring in which each color appears at each vertex v at most f(v) times where f(v) is a positive integer assigned to v. This paper gives efficient sequential and parallel algorithms to find ordinary edge-colorings and f -colorings for various classes of graphs such as bi...

A set S ⊆ V is independent in a graph G = (V, E) if no two vertices from S are adjacent. The independence number α(G) is the cardinality of a maximum independent set, while μ(G) is the size of a maximum matching in G. If α(G) + μ(G) = |V |, then G is called a König-Egerváry graph. The number dc (G) = max{|A| − |N (A)| : A ⊆ V } is called the critical difference of G [21], where N (A) = {v : v ∈...

Random graphs underly processes such as polymerization [1], percolation [2], and the formation of social networks [3, 4]. Random graphs have been extensively studied, especially in theoretical computer science [5, 6]. Special families of random graphs have been also examined, e.g., planar random graphs appear in combinatorics [7, 8] and in physics [9]. The basic framework for generic random gra...