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Article
Degree Powers in \(C_5\) -Free Graphs
Let \(G\) G be ...
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Article
Note on the Hardness of Rainbow Connections for Planar and Line Graphs
An edge-colored graph \(G\) G ...
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Article
Rainbow Connection in 3-Connected Graphs
An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rc(G), is the smallest n...
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Article
Rainbow Connections of Graphs: A Survey
The concept of rainbow connection was introduced by Chartrand et al. [14] in 2008. It is interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most ...
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Book
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Chapter
Introduction
Let G be a finite and undirected simple graph, with vertex set V (G) and edge set E(G). The number of vertices of G is n, and its vertices are labeled by v 1,v 2,…,v ...
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Chapter
The Coulson Integral Formula
In the theory of graph energy, the so-called Coulson integral formula (3.1) plays an outstanding role. This formula was obtained by Charles Coulson as early as 1940 [73] and reads:
3.1 ... -
Chapter
Bounds for the Energy of Graphs
A graph G of order n and size m is called an (n, m)-graph. In what follows we assume that the graph eigenvalues are labeled in a nonincreasing manner, i.e., λ1≥λ2≥⋯≥λ n . If G is connect...
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Chapter
Other Graph Energies
Motivated by the large number of interesting and nontrivial mathematical results that have been obtained for the graph energy, several other energy-like quantities were proposed and studied in the recent mathe...
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Chapter
Graphs Extremal with Regard to Energy
One of the fundamental questions that is encountered in the study of graph energy is which graphs (from a given class) have greatest and smallest $$\...
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Chapter
Hypoenergetic and Strongly Hypoenergetic Graphs
A graph on n vertices, whose energy is less than n, i.e., ℰ(G) < n, is said to be hypoenergetic. Graphs for which ℰ(G) ≥ n are said to be nonhypoenergetic. In [441], a strongly hypoenergetic graph is defined to b...
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Chapter
The Chemical Connection
Research on what we call the energy of a graph can be traced back to the 1940s or even to the 1930s. In the 1930s, the German scholar Erich Hückel put forward a method for finding approximate solutions of the Sch...
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Chapter
Common Proof Methods
After the concept of graph energy was proposed [149], there was much research on this topic. One basic problem is to find the extremal values or the best bounds for the energy within some special classes of gr...
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Chapter
Miscellaneous
In this chapter we have collected results on graph energy that could not be outlined elsewhere.
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Chapter
The Energy of Random Graphs
In the previous chapter, several lower and upper bounds have been established for various classes of graphs, among which bipartite graphs are of particular interest. But only a few graphs attain the equalities...
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Chapter
Hyperenergetic and Equienergetic Graphs
The energy of the n-vertex complete graph K n is equal to 2(n − 1). We call an n-vertex graph Ghyperenergetic if ℰ(G) > 2(n − 1). From Theorem 5.24, we know that for ...
