| One of the most remarkable chemical applications of graph theory is based on the close correspondence between the graph eigenvalues and the molecular orbital energy levels ofπ-electrons in conjugated hydrocarbons. For the majority of conjugated hydrocarbons, in the Huchkel molecular orbital approximation, the total 7r-electron energy E satisfies the relation: whereλi are eigenvalues of the molecular graph G representing the conjugated hydrocarbon. The right-hand sige of Eq. (0.0.2) is well-defined for all graphs. In view of this, if G is an any graph, then by means of Eq. (0.0.2) one defines E(G) and calls it the energy of the graph G.The problem of characterizing (molecular) graphs for which the condition E(G)> n is obeyed seems to be first time considered by theoretical chemists England and Ruedenberg. In 1973 they published a paper in which they asked, translated into the language of graph spectral theory, why the graph energy exceeds the number of vertices, understanding that the graph in question is a molecular graph, i. e., a connected graph with maximum degree at most 3.An n-vertex graph G is said to be hypoenergetic if E(G)< n and strongly hypoenergetic if E(G) n have been characterized. In 2007, Nikiforov showed that for almost all graphs, E=(4/3π+o(1)) n3/2. Thus the number of hypoenergetic graphs is relatively small, and then finding (strongly) hypoenergetic graphs is a meaningful topic.Let G be a graph with n vertices and m edges. The cyclomatic number of a connected graph G is defined as c(G)=m-n+1. A graph G with c(G)=k is called aκ-cyclic graph. In particular, for c(G)=0,1,2 or 3 we call G a tree, unicyclic, bicyclic or tricyclic graph, respectively.Denote by Sn the star of order n. Let W be the tree obtained from S3 by attaching two pendent edges to each of the two leaf vertices of S3. Let Q be the tree obtained from S2 by attaching two pendent edges to each of the vertices of S2. In 2008, Gutman et al. proved that S1, S3,S4 and W are the only four hypoenergetic trees with maximum degree at most 3.In Chapter 2 of this thesis, we characterize all connected graphs with maxi-mum degree at most 3 whose energies less than or equal to the number of vertices. Specifically, S1, S3, S4, W and complete bipartite graph K2,3 are the only five hy-poenergetic connected graphs with maximum degree at most 3; S2, Q, K2,2 and K3,3 are the only four connected graphs with maximum degree at most 3 whose energies equal to the number of vertices. By this, the validity of a conjecture by Majstorovic et al. has been confirmed.In Chapter 3, we consider hypoenergetic and strongly hypoenergeticκ-cyclic graphs. We first consider hypoenergetic and strongly hypoenergetic trees. For any given n and△, the existence of both hypoenergetic and strongly hypoener-getic trees of order n and maximum degree△is completely characterized. Then we show that for any givenκ≥1 there exist hypoenergetic and strongly hypoen-ergeticκ-cyclic graphs of order n and maximum degree△for all (suitable large) n and△. Finally we show that for△≥4 there exist hypoenergetic unicyclic, bicyclic and tricyclic graphs of order n and maximum degree△for all n except very few small values of n. These results greatly extend the known results for hypoenergetic graphs.Let a, b and c be integers,1≤a |
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