# Energy of Graphs

## Ghorbani, Ebrahim | 2010

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1. Type of Document: Ph.D. Dissertation
2. Language: Farsi
3. Document No: 40238 (02)
4. University: Sharif University of Technology
5. Department: Mathematical Sciences
7. Abstract:
8. Let G be a graph with adjacency matrix A and Δ be a diagonal matrix whose diagonal entries are the degree sequence of G. Then the matrices L = Δ− A and Q = Δ+A are called Laplacian matrix and signless Laplacian matrix of G, respectively. The eigenvalues of A, L, and Q are arranged decreasingly and denoted by λ1 ≥ · · · ≥ λn, μ1 ≥ · · · ≥ μn ≥ 0, and q1 ≥ · · · ≥ qn ≥ 0, respectively. The energy of a graph G is defined as E(G) :=
n
i=1
|λi|.
Furthermore, the incidence energy, the signed incidence energy, and the H¨uckel energy of G are
defined as
IE(G) :=
n
i=1

qi, LE(G) :=
n
i=1

μi, HE(G) :=

2
r
i=1 λi, n= 2r;
2
r
i=1 λi + λr+1, n= 2r + 1;
respectively. In this thesis we study these four kinds of graph energies. For chromatic number and list chromatic number of a graph, we obtain the upper bound E(G) ≥ 2max(n − χ(G), ch(G)) where χ(G) is the chromatic number of the complement of G and ch(G) is the list chromatic
number of G. The effect of removing or adding an edge on increasing or decreasing of energy is investigated. We show that for a graph G of order n with m edges, where n is even, the following holds:
HE(G) ≤
⎧⎨

2m
n−1 +

2m(n−2)(n2−n−2m)
n−1 , m≤ n3
2(n+2) ;
2
n

mn(n2 − 2m) < 4m
n, m>n3
2(n+2) .
Equality is attained if and only if G is a strongly regular graph with parameters (n, k, λ, μ) = (4t2+4t+2, 2t2+3t+1, t2+2t, t2+2t+1). A similar bound also holds if n is odd. Using properties of elementary symmetric functions, we prove that for any 0 < α ≤ 1, qα
1 +· · ·+qα
n
≥ μα1
+· · ·+μαn
.
If we let α =
in this inequality, the inequality IE(G) ≥ LE(G) follows. Using a different method, we extend above inequality to the values α ∈ (0, 1] ∪ [2, 3]. Moreover it is shown that the reverse of the inequality holds for α ∈ (1, 2). Further we show that equality holds in all of the above inequalities (excpet for α = 1, 2, 3) if and only if G is bipartite. Finally some lower bounds for IE and LE are found.
9. Keywords:

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