3 metric matrix T and r ∈ N + [19] which is equal to the row sum ∑ N j $_{=1}$T$_{ij}$ , since T T = T . We also define the sum of all entries (SE) of T as Note that SE ( T ) / N is the average row (= column) sum. Next, we observe that The lower bound is then where To obtain our lower bound, we then use the r = 1 special case of the theorem that for a non-negative sym- λ$_{max}$ ( T ) ≥ [ SE ( T $^{r}$) N ] 1 /r . (2.12) W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{BC}$$_{z}$ , q ) ≥ W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{BC}$$_{z}$ , q )$_{ℓ}$ (2.13) SE ( T ) = N ∑ i,j =1 T$_{ij}$ . (2.10) SE ( T ) = P ( sc [2$_{F}$ × ( L$_{y}$ )$_{BCy}$ × ( L$_{z}$ )$_{BC}$$_{z}$] , q ) . (2.11) W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{BC}$$_{z}$ , q )$_{ℓ}$ = lim L$_{y}$ →∞ ( SE ( T ) N ) 1 / ( L$_{y}$L$_{z}$ ) = lim L$_{y}$ →∞ [ P ( sc [2$_{F}$ × ( L$_{y}$ )$_{BCy}$ × ( L$_{z}$ )$_{BC}$$_{z}$] , q ) P ( sq [( L$_{y}$ )$_{BCy}$ × ( L$_{z}$ )$_{BC}$$_{z}$] , q ) ] 1 / ( L$_{y}$L$_{z}$ ) = [ λ$_{tube}$$_{((}$$_{L}$$_{z}$$_{)}$$_{BC}$$_{z}$$_{)}$$_{,max}$ λ$_{ts}$$_{((}$$_{L}$$_{z}$$_{)}$$_{BC}$$_{z}$$_{)}$$_{,max}$ ] 1 /L$_{z}$ . (2.14) λ$_{tube}$$_{(2)}$$_{,max}$ = 1 2 [ q 4 − 8 q 3 + 29 q 2 − 55 q + 46 + √ R$_{22}$ ] (3.4) R$_{22}$ = q 8 − 16 q 7 + 118 q 6 − 526 q 5 + 1569 q 4 − 3250 q 3 + 4617 q 2 − 4136 q + 1776 . (3.5) λ$_{ts}$$_{(2)}$$_{,max}$ = q 2 − 3 q + 3 . (3.2) be where We then substitute these results for λ$_{ts}$$_{(2)}$$_{,max}$ and λ$_{tube}$$_{(2)}$$_{,max}$ into the L$_{z}$ = 2 special case of (2.14) to obtain W ( S$_{2}$, q )$_{ℓ}$, and thus the resultant lower bound on W ( S$_{2}$$_{F}$ , q ) = W ( S$_{2}$$_{P}$ , q ) ≡ W ( S$_{2}$, q ): W ( S$_{2}$, q ) ≥ W ( S$_{2}$, q )$_{ℓ}$. We now evaluate our general lower bound in Eqs. (2.13) and (2.14) for a slab of the simple cubic lattice with thickness L$_{z}$ = 2 and FBC$_{z}$, denoted S$_{2}$$_{F}$ . In this case the transverse slice is the graph sq [2$_{F}$ × ( L$_{y}$ )$_{BCy}$]. For FBC$_{y}$, an elementary calculation yields with a single λ$_{ts}$$_{(2}$$_{F}$$_{)}$ = λ$_{ts}$$_{(2}$$_{P}$$_{)}$ ≡ λ$_{ts}$$_{(2)}$ , and this is also the maximal λ for PBC$_{y}$ [10, 20], so that III. RESULTS FOR SLAB OF THICKNESS L z = 2 WITH FBC$_{z}$ IV. COMPARISON WITH LARGE- q SERIES EXPANSIONS P ( sq [2$_{F}$ × ( L$_{y}$ )$_{F}$ ] , q ) = q ( q − 1)( q 2 − 3 q + 3) L$_{y}$ $^{−}$1 (3.1) P ( sc [2$_{F}$ × ( L$_{y}$ )$_{F}$ × 2$_{F}$ ] , q ) = P ( sc [2$_{F}$ × 2$_{F}$ × ( L$_{y}$ )$_{F}$ ] , q ) = P ( sq [4$_{P}$ × ( L$_{y}$ )$_{F}$ ] , q ) (3.3) One way to elucidate how this lower bound W ( S$_{2}$, q )$_{ℓ}$ compares with the exact W ( sq, q ) and W ( sc, q ) is to com- pare the large- q series expansions for these three func- tions. For this purpose, it is first appropriate to give some relevant background on large- q series expansions for W ( { G } , q ) functions. Since there are q n possible color- ings of the vertices of an n -vertex graph G with q colors We next use the calculation of in Ref. [34] (where each of the 2$_{F}$ BC’s is equivalent to 2$_{P}$ ), from which we calculate the maximal λ$_{tube}$$_{(2)}$$_{,max}$ to