enumeration of the number of spanning trees in some ... - Toubkal

CHAPTER 5.90THE NUMBER OF SPANNING TREES OF CERTAIN FAMILIES OFPLANAR MAPSThe characteristic equation is r 2 − hr + 1 = 0, ∆ = h 2 − 4, if h ≥ 2 ⇒ ∆ ≥ 0 thereforethe solutions are: r 1 = h−√ h 2 −4and r2 2 = h+√ h 2 −4, hence:2τ(C n ) = α( h − √ h 2 − 42) n + β( h + √ h 2 − 4) n , α, β ∈ R, and n ≥ 12Using the initial condition τ(C 1 ) = h, τ(C 2 ) = h 2 − 1, we obtain:α = (h+√ h 2 −4)(−h 2 +2+h √ h 2 −4)4 √ h 2 −4, β = (h−√ h 2 −4)(h 2 −2+h √ h 2 −4)4 √ , hence the result.h 2 −4□Remark 5.5.3 In Theorem 5.5.1, if we replace k = 1, we obtain the sequence of mapsC n shown in Figure 5.6, thenτ(C n ) =(1√h2 − 4( h + √ h 2 − 425.6 Other values of k i = 1 and h i = h) n+1 − ( h − √ h 2 − 4)), n+1 n ≥ 1.2In this section, we fix the value of k i by 1 and change the value of h i ; subsequently, wederive a simple formula for the number of spanning trees of special families of planarmaps (classical results), called the n-Fan chains, the n-Grid chains, the n-Tent chains,the n-Hexagonal chains and the n-Eight chains, ... etc.• The n-Fan chains (k = 1 and h = 3)If we take h = 3 in the sequence of maps C n in Figure 5.6, we obtain the sequencen-Fan chains F n , where n is the number of triangles (|V Fn | = n + 2) [39]; (seeFigure 5.7).Figure 5.7: The n-Fan chains F n