Some Extremal Problems About The Largest Eigenvalue Of Polyominos And Hexagonal Chains

Extremal polyomino chains and hexagonal chains concerning largest eigenvalue are studied in this thesis.A polyomino chain is a kind of polyomino systems, in which inner surfaces are regular squares of unit edge length and each vertex belongs to three squares at most. Denotedbyψ_n the set of the polyomino chain with n squares and L'_n the linear chain inψ_n.A hexagonal chain is a kind of hexagonal system, in which inner surfaces are regular hexagons of unit edge length, each vertex belongs to two hexagons at most and each hexagonis adjacent to two hexagons at most. Denoted byΓ_n the set of the hexagonal chain with n hexagons, L_n the linear chain and H_n the helicene chain inΓ_n .The largest eigenvalue of Gis just the largest eigenvalue of the characteristic polynomial of G and denoted by ,x₁ (G).There are four chapters. Related backgrounds and results are discussed in the first chapter. Some related terms and important results are introduced in the second chapter; Aconclusion is verified in the third chapter that for (?)n≥1 and (?)B_n∈Ψ_n, if B_n≠L'_n, then x₁(B_n)>x₁(L'_n). The scope of the chain with second maximal (or minimum) largest eigenvalue is given in the fourth chapter: denoted by D_n∈Γ_n-{H_n}and B_n∈Γ_n-{H_n,D_n}, if x₁(B_n)1(D_n) for (?)n≥4, then D_n∈Φ_n); denoted by D_n∈Γ_n-{L_n} and B_n∈Γ_n-{L_n,D_n}, if x₁(B_n)>x₁(D_n) for (?)n≥4, then D_n∈Ω_n. whereand k_i is the way to attach the (i+1)th hexagon to D_i=βk₂……k_i-1, according to threedifferent positions of attaching, we call itα,βandγfrom the top down.