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Energy of Graphs
, Ph.D. Dissertation Sharif University of Technology ; Akbari, Saeid (Supervisor)
Abstract
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=...
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=...