The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let Gn,m be the set of connected graphs of order n and with m edges. In this note we determined the extremal graphs from Gn,m with n ≤ m ≤ 2n−4 minimizing the matching energy. Also we determined the minimal matching energy of graphs from Gn,m where m = n−1+t and 1 ≤ t ≤ β−1 and w...

Extremal problems for graph homomorphisms have recently become a topic of much research. Let hom(G,H) denote the number of homomorphisms from G to H. A natural set of problems arises when we fix an image graph H and determine which graph(s) G on n vertices and m edges maximize hom(G,H). We prove that if H is loop-threshold, then, for every n and m, there is a threshold graph G with n vertices a...

Let I(G) be a topological index of a graph. If I(G+ e) < I(G) (or I(G+ e) > I(G), respectively) for each edge e ∈ G, then I(G) decreases (or increases, respectively) with addition of edges. In this paper, we determine the extremal values of some topological indices which decrease or increase with addition of edges, and characterize the corresponding extremal graphs in bipartite graphs with a gi...

The extremal characteristics of random structures, including trees, graphs, and networks, are discussed. A statistical physics approach is employed in which extremal properties are obtained through suitably defined rate equations. A variety of unusual time dependences and system-size dependences for basic extremal properties are obtained.

A graph G is said to be F-residual if for every point u in G, the graph obtained by removing the closed neighborhood of u from G is isomorphic to F. Similarly, if the remove of m consecutive closed neighborhoods yields Kn, then G is called m-Kn-residual graph. Erdös determine the minimum order of the m-Kn-residual graph for all m and n, the minimum order of the connected Kn-residual graph is fo...

Given a family of graphs H, the extremal number ex(n,H) is the largest m for which there exists a graph with n vertices and m edges containing no graph from the family H as a subgraph. We show that for every rational number r between 1 and 2, there is a family of graphs Hr such that ex(n,Hr) = Θ(n r). This solves a longstanding problem in the area of extremal graph theory.

Let G be a graph of order n with Laplacian spectrum μ1 ≥ μ2 ≥ · · · ≥ μn. The Laplacian-energy-like invariant of graph G, LEL for short, is defined as: LEL(G) = n−1 ∑ k=1 √ μk . In this note, the extremal (maximal and minimal) LEL among all the connected graphs with given matching number is determined. The corresponding extremal graphs are completely characterized with respect to LEL. Moreover ...

In this paper, we study the signless Laplacian spectral radius of bicyclic graphs with given number of pendant vertices and characterize the extremal graphs.

Let G be a graph of order n with Laplacian spectrum μ1 ≥ μ2 ≥ · · · ≥ μn. The Laplacian-energy-like invariant of graph G, LEL for short, is defined as: LEL(G) = n−1 ∑ k=1 √ μk . In this note, the extremal (maximal and minimal) LEL among all the connected graphs with given matching number is determined. The corresponding extremal graphs are completely characterized with respect to LEL. Moreover ...