Bounds for the Energy of Graphs

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Aashtab, Aarman | 2022

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Type of Document: Ph.D. Dissertation
Language: Farsi
Document No: 55578 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Akbari, Saieed
Abstract:
The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E(G) ⩾ n. Here, we improve this result by showing that if G is a connected subcubic graph of order n, n ⩾ 8, then E(G) ⩾ 1.01n. Also, we prove that if G is a traceable subcubic graph of order n,then E(G) ⩾ 1.1n. Let G be a connected cubic graph of order n, it is shown that E(G) > n + 2, for n ⩾ 8 and we introduce an infinite family of connected cubic graphs whose for each element, say G, E(G) ⩾ 1.24n, and some important conjectures will be raised about this. At the end, for a graph G and its vertex induced subgraphs H and K, whose vertices partition the vertices of G, we will check the equality conditions of E(G) = E(H) + E(K) and for this case, two important inequalities rank(G) ⩽ rank(H) + rank(K) and ρ(G) ⩽ ρ(H) + ρ(K) will be proved.
Keywords:
Energy ; Regular Graph ; Induced Subgraph ; Subcubic Graph ; Spectral Radius ; Graph Energy