ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 379-408 Spectra and structural polynomials of graphs of relevance to the theory of molecular conduction Patrick W. Fowler *, Barry T. Pickup Department of Chemistry, University of Sheffield, Sheffield, S3 7HF, UK Irene Sciriha Department of Mathematics, University of Malta, Msida, Malta Martha Borg Department of Chemistry, University of Sheffield, Sheffield, S3 7HF, UK Received 3 November 2016, accepted 28 February 2017, published online 24 March 2017 In chemistry and physics, distortivity of n-systems (stabilisation of bond-alternated structures) is an important factor in the calculation of geometric, energetic, and electronic properties of molecules via graph theoretical methods. We use the spectra of paths and cycles with alternating vertex and edge weights to obtain the eigenvalues and eigenvectors for a class of linear and cyclic ladders with alternating rung and backbone edge weights. We derive characteristic polynomials and other structural polynomials formed from the cofactors of the characteristic matrix for these graphs. We also obtain spectra and structural polynomials for ladders with flipped weights and/or Möbius topology. In all cases, the structural polynomials for the composite graphs are expressed in terms of products of polynomials for graphs of half order. This form of the expressions allows global deductions about the transmission spectra of molecular devices in the graph-theoretical theory of ballistic molecular conduction. Keywords: Adjacency matrix, characteristic polynomial, molecular conduction, eigenvalues, weighted graphs. Math. Subj. Class.: 05C50 * Corresponding author. E-mail addresses: p.w.fowler@sheffield.ac.uk (Patrick W. Fowler), b.t.pickup@sheffield.ac.uk (Barry T. Pickup), irene.sciriha-aquilina@um.edu.mt (Irene Sciriha), mborg1@sheffield.ac.uk (Martha Borg) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 380 Ars Math. Contemp. 13 (2017) 275-291 1 Introduction Our aim is to use weighting of graphs as a tool for the study of ballistic molecular conduction in undistorted and distorted molecular and extended systems. In this article we derive the spectra and characteristic polynomials of a series of graphs that possess three common features. The first is that they are bipartite. The second is that they possess an involution that allows the graph to be expressed as a product of simpler graphs with known spectra. The third feature is termed 'distortivity' by physical scientists. This refers to the way that the spectrum changes with edge weights, and is of prime importance in theories of electronic structure, where molecular structures are modelled by graphs. It is well known to physicists and chemists that extended overlapping n-electron systems may achieve greater stability by distorting in such a way that bond lengths alternate, and the sharing of electron density across the n-system is reduced. This is known in the physics literature as Peierls distortion [13], and in the chemical literature as Jahn-Teller distortion [12]. It typically affects n-electron systems in such a way as to reduce their conductivity. In order to assess the importance of distortivity for the specific phenomenon of ballistic molecular conduction, we need explicit characteristic polynomials and spectra for families of weighted graphs representing molecules of chemical interest. 1.1 Graph theoretical background The graphs in which we are interested are linear ladders, their cyclic analogues the treadmills, and graphs derivable from them by using (signed or zero) weights, such as linear polyacenes and (Mobius) cyclacenes, shown in Fig. 1. In graph theory terms, we can mimic (a) ra:m (b) (c) cco " :cooc Figure 1: Families of graphs treated in this paper: (a) ladders; (b) treadmills; (c) linear polyacenes; (d) cyclic polyacenes. geometric distortion of a molecular framework by studying weighted graphs in which edge weights alternate [11]. Adjacency matrices of such graphs have been studied by Gover [9] in the form of 2-Toeplitz matrices. Gover gave an explicit solution for the spectra of 2-Toeplitz matrices of odd dimension, and an implicit solution for even dimensions. These solutions form the basis for our treatment of ladders and treadmills. Ladders (treadmills) comprise two backbone chains (rings), that are linked by 'rungs'. We shall alternate the weights on the rungs, and separately on the edges comprising the two backbones, in such a way that an involution symmetry is preserved. This symmetry element swaps vertices in upper and lower backbone chains of the graphs and is crucial for the solution of the secular problem for distorted and undistorted systems. The use of symmetry splits the secular matrices of the graphs into two non-interacting blocks, each of which represents a single P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 381 path (or cycle) with alternating weighted vertices and edges. It is these backbone graphs that possess the analytical solutions derived previously by Gover [9] and Shin [18]. An active avenue of research is exploration of the influence of molecular topology in conduction behaviour. We therefore include certain graphs with edge weights having flipped signs, and/or with a pair of crossed backbone edges. These flipped and crossed graphs are sometimes called Möbius graphs [6]. The cases that we consider here have closed-form spectra and structural polynomials that can be derived using the methodology used for unflipped, uncrossed graphs. 1.2 Physical motivation The physical context for the present mathematical exploration is that electronic structure of unsaturated carbon networks is qualitatively modelled using spectral graph theory. In particular, the basic reason for our interest in the graphs described in this paper is our research into molecular conductivity in small molecules [14, 15] using the source-sink-potential (SSP) method of Ernzerhof et al. [3,4, 5, 20]. This approach uses graph theory as a vehicle for showing important qualitative features in electron transmission for individual molecules. Central to the SSP method is the idea of a molecular device based on a molecular graph, in which the effects of infinite attached wires are represented by two special extra vertices, which behave respectively as a source and sink (of electrons). We have shown [14] that electronic transmission in this model can be expressed using a basic set of polynomials related to the molecular graph, G. These are the characteristic polynomial, s(E) = det(E1 - A), (1.1) and the cofactors of the characteristic matrix, .WE) = (-1)p+q det(E1 - A)[p'q] = (El - A)-q1s(E), (1.2) where E is the energy of the transmitted electron, 1 is the n x n unit matrix and A is the n x n adjacency matrix of the graph G of order n. The indices in square-brackets refer to the sets of rows and columns deleted from the determinant of the characteristic matrix. The eigenvalue problem, Ack = ckek, (1.3) allows us to define the n eigenvalues {ek}, and the corresponding eigenvectors ck. Spectral decomposition allows us to write n s(E) = n (E - ek), (1.4) k=1 and spectral resolution of the inverse gives a general expression for all .pq(E) polynomials in terms of eigenvectors and eigenvalues of A: n .pq(E ) = E f-^ <'-5> k=1 where cpk is the pth entry in the kth eigenvector, ck. In what follows we will find it useful to switch between the two approaches, viz. calculating structural polynomials from 382 Ars Math. Contemp. 13 (2017) 275-291 determinants of characteristic matrices, or from explicit solutions of the eigenvalue problem Eq. (1.3). For a specific device in which vertices p and q of the molecular graph are attached to infinite conducting wires, one needs just four polynomials, namely, s, -pp, -qq, and Vpq,pq(E) = det(E1 - A)[pq,pq] (1.6) to deduce an expression for the transmission, T(E), of an incoming stream of electrons [14]. These are all characteristic polynomials derived from vertex-deleted graphs: s = p(G ,E), t = Jpp = p(G - P, E), u = Jqq = P(G - q, E), V = Vpq,pq(E)= p(<- p - q,E), (1.7) where p(<, E) is the characteristic polynomial of graph <, and the letters s, t, u, and v refer to literature notation [14]. The formula for vpq pq can be deduced using Jacobi's relation [19]: svpq,pq = -pp-qq — -pq. (1.8) For convenience, we refer to s(E), the -pq(E) and v(E) as structural polynomials. We have shown [15] that molecular conduction can be thought of in two different ways, i.e. either as occurring through molecular bonds (graph edges), or through individual molecular orbitals (eigenvectors of the graph adjacency matrix). We find that there are 11 basic categories of conduction [7, 15, 17] for molecules and that these are determined by the eigenvector coefficients. Conduction behaviour at eigenvalues of the adjacency matrix is particularly important [15]. Hence arises our interest in closed-form expressions for spectra and structural polynomials. Spectral representations of the structural polynomials are also informative, in that they allow elaboration of the SSP model to treat the physically important effects of Pauli exclusion, an effect that prevents current passing through filled orbitals. This extension of the theory is worked out in a recent paper [16]. We can summarise the key features of our approach and the main results as follows. Explicit expressions for structural polynomials, spectra and eigenvectors of weighted paths and cycles are obtained. These are useful in themselves for the discussion of distortivity and conduction. We then exploit the graph-product structure of the families of ladders, treadmills and Möbius forms to build analytical expressions for the structural polynomials and spectral properties of these graphs in terms of those of the simpler graphs. This gives compact formulas that are ultimately related to Chebyshev and similar orthogonal polynomials. It is this 'factorised' form of the final expressions that gives a powerful tool for interpretation of spectra and conduction properties of ladders, treadmills. This interpretation will be used to analyse the effects of flips and twists on conduction in physically realisable systems. 1.3 Plan The plan of the paper is as follows. First we derive eigenvectors and eigenvalues for weighted alternating paths (Section 2), and then derive expressions for the important structural polynomials in Section 3. These results are used to derive spectra for ladders and P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 383 their structural polynomials in Sections 4 and 5. The spectra for alternating cycles are derived in Section 6, and their structural polynomials in Section 7. Derivations of spectra and structural polynomials of treadmills then follow in Sections 8 and 9. Section 10 introduces important chemical graphs that can be derived from ladders and treadmills. We end with a brief conclusion. Our explicit treatment of cases necessarily leads to a large number of equations, but the central results are Eqs. (5.6) and (9.3), which show the generic relationships between the structural polynomials of ladders and chains, and treadmills and cycles, respectively. The structural polynomials for weighted chains are given in Eqs. (3.12) and (3.13), and weighted cycles in Eqs. (7.13) and (7.14) and for flipped cycles in Eqs. (7.15) and (7.16). The blocks of equations giving the results are: Eqs. (5.7) to (5.10) for ladders; Eqs. (9.4) and (9.5) for treadmills; Eqs. (9.9) and (9.10) for flipped treadmills; Eqs. (9.14) and (9.15) for Möbius treadmills; Eqs. (9.19) and (9.20) for flipped Möbius treadmills. 2 The spectra of alternating weighted paths PM (a, b | c,d) We consider paths, PM(a, b | c, d), with alternating vertex weights a, b, and edge weights c, d. Eigenvalues and eigenvectors for such weighted paths have been deduced by Gover [9] and Shin [18]. Gover used recursion to show that the spectrum of the odd-vertex chain, P2N+1, could be expressed in terms of two sets of polynomials. One is the Chebyshev polynomials of the second kind, UN. The other set of polynomials satisfy the Chebyshev recursion relation, but with different initial values. The eigenvectors for the odd paths are evaluated at the zeroes of the polynomial UN. The even paths have an analogous form for eigenvectors and eigenvalues, but one of the quantities cannot be evaluated analytically. We discuss odd and even paths separately. 2.1 The odd path, P2N+1 (a, b | c, d) A path, P2N+1 (a, b | c,d), with 2N +1 vertices is shown in Fig. 2. It is convenient to write Figure 2: A chain, P2N+i(a b | c, d), with 2N + 1 vertices and alternating vertex weights (a, b), and edge weights (c, d). the adjacency matrix, AP, for this bipartite graph in the form 1 c 2 d 3 c...... d 2N-1 c 2N d 2N+1 ababbaba (2.1) where 1h symbolises a unit matrix of dimension h, and superscript T indicates a transpose. We place the (N + 1) odd-numbered vertices shown in Fig. 2 in the first block, and the N 384 Ars Math. Contemp. 13 (2017) 275-291 even-numbered vertices in the second. The (N +1) x N-dimensional matrix BP is then BP c 0 • • 0 0 d c 0 0 0 0 .d c 0 0 • • 0 d (2.2) In order to find the eigenvalues of the matrix AP we can use the fact that the blocks on the diagonal are invariant to any unitary transformation. Therefore, a singular value decomposition of the off diagonal block will render the whole matrix in a form in which each block is diagonal or pseudo-diagonal. This technique has been used [15], for example, to provide a compact derivation of the Coulson-Rushbrooke theorem for bipartite graphs [1]. The singular value decomposition [8, Sections 2.5.3 and 2.5.6] of BP can be written as BPXP YPaP, (2.3) where XP, and YP are N- and (N + 1)-dimensional orthogonal matrices, respectively. The (N +1) x N-dimensional rectangular matrix, ctp, is "diagonal", i.e. M 0 • • 0 \ 0 ap . .. 0 0 0 0 • P . aN • 0/ (2.4) and the singular values ap > 0, have labels k. The theory of singular value decomposition tells us further that (BP) BPX PP k Xp (ak ) BP (BP)T Yp = Yp(ap)2 with ap > 0 for k = 1,..., N, and app+1 = positive definite tridiagonal matrix for k = 1,..., N, for k = 1,..., N +1, (2.5) 0. We note that the N x N-dimensional (bp)t bp /c2 + d2 cd 0 cd c2 + d2 cd 0 cd c2 + d2 0 c2 + d2 cd 0 0 0 cd ! + d2) (2.6) 0 0 c P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 385 represents the adjacency matrix of a path of length N with equal vertex weights (c2 + d2), and equal edge weights cd, so the eigenvalues are (ap)2 = c2 + d2 + 2cdcos ûp for k = 1, 2,..., N, where the angle nk N + 1 also describes the orthonormal eigenvectors Xppk = Np sin pûp, and the normalisation factor is Np = 2 N + 1 (2.7) (2.8) (2.9) (2.10) The (N +1) x (N + 1)-dimensional semi-definite tridiagonal matrix, on the other hand, (c2 cd 0 ••• 0 0 \ cd c2 + d2 cd '.. 0 Bp (Bp) 0 0 \0 cd 0 + d2 0 .. 0 c2 + d2 cd cd d2 (2.11) has no such simple expressions for its eigenvectors, but they can be derived directly from the singular value decomposition. We can use Eq. (2.3) to deduce that 1 Yppk = — (cXpk + dXp_hk) for k, p = 1,2,..., N. (2.12) We note from Eq. (2.9) that p = 0 implies Xpk = 0, and p = N +1, implies Xp+1 k = 0. The nullspace vector is YpN+i = Np+i (-1)p-1 cp-1dN-p+1, where the normalisation factor is NN+1 d2 c2 d2N+2 _ c2N+2 ' We define the (2N + 1)-dimensional orthogonal matrix Wp Yp 0 0 Xp which gives (Wp)T ApWp al N +1 (Yp)T BpXp ' Xp T Bp T Yp bl (2.13) (2.14) (2.15) (2.16) N 2 T c k 386 Ars Math. Contemp. 13 (2017) 275-291 The off-diagonal blocks in Eq. (2.16) simplify because (yp) t bpxp = ap, (2.17) where aP is given in Eq. (2.4). It is evident that each block of (WP)T AP WP is diagonal, so that it comprises N two-dimensional interacting blocks of the form 4 Ik (2.18) and a single one-dimensional block with eigenvalue a. The two-dimensional blocks give 2N eigenvalues Ep± = + b) ± 1 Dp for k = 1, 2,..., N (2.19) with discriminant DP = y/(a - b)2 + 4(ip)2 = yj(a - b)2 + 4(c2 + d2 + 2cdcos ÛP). (2.20) The eigenvectors arising from these two-dimensional blocks can be written as NP± (epIP- a (2.21) where the normalisation constants are NP+ DP (E2+ - a) and NP DP (a - Ep_) (2.22) We can write the 2N + 1 eigenvectors of A P in the form cp± for k = 1, 2,..., N, and cp +1, the latter arising from the extra null space eigenvector in the singular value decomposition. Using expression (2.12) for YP, we obtain expressions for the eigenvector coefficients cpp,k± = Np± (Ep± - a) XpPk, cpp-1,k± = Nk± (cXpk + dXp—1,0 . (2.23) There is, in addition, a single eigenvalue arising from the one-dimensional block, and corresponding to the null-space eigenvector in the YP subspace, that is of the form EP+1 — a. (2.24) The corresponding eigenvector has coefficients c2p,N+1 — 0, c .p 2p-1,N +1 Np+1 (-1)p-1 cp-1dN-p+1 (2.25) 1 1 P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 387 2.2 The even path, P2N(a, b | c, d) The adjacency matrix for the bipartite graph, P2N(a, b | c, d), can be written as BP' AP a1N (BP) b1 N (2.26) using the same numbering scheme for vertices as that shown in Fig. 2. The adjacency matrix is identical to Eq. (2.1), but with one row missing, so that BP is a square N x N matrix. We again use singular decomposition [8, Sections 2.5.3 and 2.5.6] of BP as shown in Eq. (2.3), where XP and YP are both N-dimensional orthogonal matrices. We note that the N x N-dimensional positive definite tridiagonal matrix has the form /c2 + d2 cd 0 ••• 0 0\ cd c2 + d2 cd ... 0 (bp)t bp = 0 cd c2 + d2 ... ... 0 0 V 0 0 '.. c2 + d2 cd 0 cd c2 We introduce an Ansatz for the eigenvectors of the matrix in Eq. (2.27) as (2.27) XpPk = NkP sinpC (2.28) where Np is a normalization factor, and 0p is an angle yet to be determined. Examining the first row of (BP)T BPXP, leaving out the normalisation factor, we find (c2 + d2) sin &P + cd sin2^P = aP2 sin q, then the indices p and q are swapped on the right-hand side of Eq. (5.7). For vertices on different backbone chains, and assuming again p < q, E2 _ „2 , jLp4,J2+2 = ^^(cd)2N—^(E - a)Up— i(x)Un—q(x)UN(x) -(E + a)Un (x)Up—i(x )Un—q(x ) j, ^Lp4—!^—i = 2(cd)2N—1{ (E + a)Up—i(x; c, d)lJN +i—q(x; d,c)UN(x) - (E - a)Un(x)t/p—i(x ; c, d)iJN+i—q(x; d, c) j , E2 _ „2 , _ 1 = ^^(cd)2N— 1 {Up—i(x)Un—q(x; d,c)UN(x) - Un(x)Up—i(x)Un—q(x; d, c) j , E2 _ „2 , _ = ^^(cd)2N— 1 {Up—i(x; c, d)UN—q(x)UN(x ) - un(x)t/p—i(x ; c, d)UN—q(x)j . (5.8) If p > q, then the indices p and q are swapped on the right-hand side of Eq. (5.8). Comparing Eqs. (5.7) and (5.8), we observe a sign change in the expressions that arises from the sign patterns of the antisymmetric functions in Eq. (4.5). The structural polynomials, expressed in terms of the half-ladder for L4N(a, b | c, d), 394 Ars Math. Contemp. 13 (2017) 275-291 where both vertices are on the same backbone path and p < q, are 1 d2(cd)2N-3 {(E - a)Up-1(x)UN-q(x; c, d)UN(x; c, d) + (E + a)UN(x; c, d)Up-1(x)UN-q(x; c, d) j , 1 d2(cd)2N_3 j (E - b)Up-1(x; c, d)UN-q(x)UN(x; c, d) + (E + 6)Un(x; c, d)Up_1(X; c, d)UN-q(x)j , 1 d2(cd)2N-3{(E - a)(E - b)Up-1(x)UN-q-1 (x)UN(x; c, d) +(E + a)(E + 6)Un(x; c, d)Up-1(x)UN-q-1(x)j, 1 d2(cd)2N-3 |Up-1(x; c, d)Un-q(x; c, d)UN(x; c, d) + UN(x; c, d)Up-1(x; c, d)UN-q(x; c, d) j . (5.9) If p > q, p and q are swapped on the RHS of Eq. (5.9). For vertices on different backbone paths and p < q, .Lp4,2q = 2d2(cd)2N-3 {(E - a)Up-1(x)UN-q(x; c, d)UN(x; c, d) - (E + a)UN(x; c, d)Up-1(x)UN-q(x; c, d) j , .Lp4-1,2q-1 = 1 d2(cd)2N-3{ (E - b)Up-1(x; c, d)UN-q(x)UN(x; c, d) - (E + 6)Un(x; c, d)Up-1(x; c, d)UN-q(x)j , .Lp4,^q+1 = 1 d2(cd)2N-3{ (E - a)(E - b)Up-1(x)UN-q-1(x)UN(x; c, d) -(E + a)(E + 6)Un(x; c, d)Up-1(x)UN-q-1(x)j, .2pf-1,2q = 2 d2(cd)2N-3 { Up-1(x; c d)UN-q(x;c, d)UN(x;c, d) - UN(x; c, d)Up-1 (x; c, d)UN-q(x; c, d)j . (5.10) which exhibit the same sign change as in Eq. (5.8) for L4N+2 ladder. If p > q, then p and q are swapped on the RHS of Eq. (5.10). 6 Alternating cycles We restrict our attention to even cycles with alternating weights. We consider separately the standard cycle, C2N(a, b | c, d), and the flipped cycle, CfN(a, b | c, d). 6.1 Alternating cycles, C2N(a, b | c,d) The 2N-vertex cycle, C2N(a, b | c, d), has alternating weights as displayed in Fig. 4. The quantities (a, b) are in this case vertex weights, and (c, d) are edge weights. It is convenient .L4 N .2p,2q .2p-1,2q-1 .L4N .2p,2q+1 2 4N P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 395 2N-2 2N-1 2N 1 Figure 4: A ring, C2N(a, b | c, d), with 2N vertices and alternating vertex weights (a, b) and edge weights (c, d). to write the adjacency matrix for this bipartite graph as a1N BC .c _ I alN) ,(BC)T blN (6.1) where we have placed the N odd-numbered vertices in the first block, and the N even-numbered vertices in the second. The N x N matrix BC is BC (c 0 • • • 0 d\ d c ... 0 0 ... ... ... . 0 .. .. 0 0 d c) (6.2) BC is the adjacency matrix of a directed, weighted N-cycle. It is easy to show that where with BCXC = XC^C, n£k' = 4k' (c + d(wC) ), w = exp 2ni The kth eigenvector has entries xCk = ^ ex^2nNr) for p, k = 1,2, We use Eq. (6.6) to define the 2N-dimensional unitary matrix w I XC 0 W =' 0 XC , N. (6.3) (6.4) (6.5) (6.6) (6.7) 396 Ars Math. Contemp. 13 (2017) 275-291 so that + C , a1N XCtBCXC WtA°W = Uct (IBC)T XC bl (6.8) where the f sign denotes the Hermitian conjugate. This transformation achieves a block diagonalization comprising N two-dimensional blocks of the form a c + d(wC)-k c + d(wC)k b each with eigenvalues EC± = x(a + b) ±- DC for k=1,...,N, where the discriminant is DC = \ (a - b)2 + 4 c2 + d2 +2cdcos /2nk\ (6.9) (6.10) (6.11) The eigenvectors can be written as entries in the vector cC: c2p-1,k± = Nk±XPk (Ek± - b) , cCp,k± = NkC±(cXpCk + dXpC+i,k). The normalisation constants are DC (EkC+ - b) and DC (b - EC-)" (6.12) (6.13) 6.2 Flipped alternating cycles, (a, b | c,d) The 2N-vertex cycle, (a, b | c, d), has alternating weights as displayed in Fig. 4, except that a single weight has a changed sign; without loss of generality, we shall flip the (1,2N) edge. It is convenient to write the adjacency matrix as AC alw BC !BC/^ blN (6.14) in the same manner as in Section 6. The N-dimensional matrix BCf is hence defined by B Cf c 0 • • 0 -d d c 0 0 0 0 .d c \0 0 • •0 d (6.15) 1 1 P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 397 Proceeding as before, BC f X C f = X C f nC f, (6.16) where fiCf/ = ¿kk'(c + d(^c/)-(2k-1)), (6.17) with ^ =exK N)> (6.18) and the eigenfunctions are XpCf = ^ exp { } for p, k = 1,2,..., N. (6.19) The derivation proceeds exactly as for the simple cycle. The eigenvalues are f 1 1 f EC± = 2 (a + b) ± ^ DC for k = 1,...,N (6.20) and the discriminant is DC = y (a - b)2 + 4 ^c2 + d2 + 2cdcos ^^ ) . (6.21) The eigenvectors are written as entries in the vector cCf as cCp-i,k± = NCf XpCk (Ed - b) , cCpf,k± = NCi (cXpCkf + dXpC+ i,k), (6.22) which is the analogue of Eq. (6.12), and the normalisation constants are NC+ = V DC' (Eg - a) and NC- = \/ Df (a- EC-)' 7 Structural polynomials of alternating cycles We derive expressions for the structural polynomials of even cycles and flipped cycles in this section. The characteristic polynomial, s°2N (E), for the graph C2N(a, b | c, d) is N sC2N = n(E - EC+)(E - EC-) k=1 n / 2 k \ = n ((E - a)(E - b) - c2 - d2 - 2cdcos j . (7.1) k=1 ' Expressing the cosine in terms of the half-angle, we find that N k=1 2N N sC2N = (4cd)^ ( (E - a)(E "bJ - (c - d)2 - cos2 g^ (4cd)N n {y - cos (7.2) k=1 ^ ' 398 Ars Math. Contemp. 13 (2017) 275-291 2Tn (z) - 2= ] ( - 2 cos ' (7'3) where y is defined in Eq. (3.10). We use the well-known relation N I 2z _ 2 cos . , k=1 x 7 where the Chebyshev polynomial of the first kind is TN (cos 0) = cos(n0). We conclude that scn = 2(cd)N {T2n (y) - 1} . (7.4) We can also derive a formula for the half ring by using Eq. (7.4) in conjunction with the standard formula T2N(z) = Tn (2z2 - 1)' (7.5) to give sc2N = 2(cd)N (Tn (x) - 1), (7.6) where we have used the definition of x in Eq. (3.5). 2N( NN The characteristic polynomial, sCfN (E), for the graph CfN(a, b | c, d) is scfN = ](E - Eg)(E - EC-) = I] ("cdx2 - 2cdcos . (7.7) k=1 k=1 ^ ' (4cd)N 1 (y2 - cos2 n(2" = 4(-cd)NTn(y)TN(-y), (7.8) V—1 v ' Expressing the cosine in terms of the half-angle, we find that N sCfN = (4cd)N ] ( y2 - cos2 "(2_- 1M = 4(-cd)Ni n( k=1 where we have used y as in Eq. (3.10), and the well-known relation Tn(z) = 2n-1II {z - cos (^^)} . (7.9) The product formula 2TN (z) = T2n (z) + 1 (7.10) and the parity of the Chebychev polynomials gives the final 'full-ring' expression scfN =2(cd)N (T2n(y) + 1). (7.11) We can also derive a formula for the half ring using the transformation in Eq. (7.5), to give sCfN =2(cd)N (Tn (x) + 1). (7.12) The remaining structural polynomials can be deduced using the fact that removal of a vertex from a cycle gives rise to a path, and we have already derived the characteristic polynomials of even and odd vertex paths in Section 3. There are two kinds of vertex in P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 399 our alternating cycles, odd-numbered vertices with weight a, and even-numbered vertices with weight b. When one of these vertices is removed we create a path of length 2N - 1. The equations for the diagonal parts of jpq for C2N(a, b | c,d) are: Ä = (cd)N— i(E - a)UN-1(x), •?2Cp2+i,2p+i = (cd)N— i(E - b)UN— i(x). (7.13) The formulae for the off-diagonal j polynomials can be deduced using Eq. (1.6), and by noting that if we remove a second vertex, then we form two (or possibly one) paths. This requires some trivial but lengthy trigonometry. The results are (E - a)(cd)N-i {Un—q— 1 (x) + Uq— i(x)} , (E - b)(cd)N —1 {Un—q—i(x) + Uq—1 (x)} , (cd)N—1 { Un—q—i(x; c, d) + Uq(x; d,c)} . (7.14) Note that the formulae do not depend upon p, but only upon q, the offset along the ring. The results for the flipped cycle can be calculated in the same manner. The diagonal j quantities are identical to those for the cycle. This is because the deletion of a vertex creates a chain, and flipped edges can be removed from a tree using an orthogonal transformation. Hence, J&p = (cd)N—i(E - a)UN —i(x), jS'+Up+i = (cd)N—i(E - b)UN —i(x), (7.15) and further (E - a)(cd)N—i {Un—q—i(x) - Uq—i(x)} , (E - b)(cd)N —i {Un—q—i(x) - Uq—i(x)} , (cd)N —i { Un—q—i(x; c,d) - Uq (x; d,c)} . (7.16) The changes in sign between Eqs. (7.14) and (7.16) arise because of the sign change between Eqs. (7.6) and (7.12). 8 Alternating treadmills Treadmills are cyclic ladders. We consider a 4N-vertex treadmill, T4N (a, b | c,d), with rung edge weights (a, b), and backbone weights (c, d) displayed in Fig. 5. We shall also consider a related treadmill, namely the 'flipped' treadmill, TfN (a,b | c,d), which has a pair of symmetrically related backbone edges with weights having a changed sign. We also include in our discussion the Möbius treadmill, T^N(a, b | c, d), which has a pair of crossed edges connecting top and bottom rings. There is also the flipped Möbius treadmill, TMn (a, b | c,d), which has a pair of crossed bottom and top edges with weights having a changed sign. All of these treadmills possess the same involution symmetry, and hence can be treated in the same way. JC2N _ j2p,2p+2q = C2N _ j2p+i,2p+2q+i _ j2 2N _ J2p+i,2p+2q+2 _ j2 2N j2p,2p+2q 2 j2 2N j2p+i,2p+2q+i 2 j2 2N J2p+i,2p+2q+2 400 Ars Math. Contemp. 13 (2017) 275-291 c 2N d 1 c 2 d 3 c 4 b 2N-2 d 2N-1 c a b a b c 2N d 1 c 2 d 3 c 4 2N-2 d 2N-1 c Figure 5: A treadmill, T4N(a, b | c, d), with 4N vertices and alternating rung (a, b) and ring edge weights (c, d). b a 8.1 The treadmill T4N(a, b | c,d) This system has an involution symmetry based upon exchange of vertices attached to the ends of rungs. The methodology proceeds in exactly the same way as for the ladder example in Section 4. Hence, the 4 x 4 blocks of vertices (top ring odd, top ring even, bottom ring odd, and bottom ring even) produce the adjacency matrix (8.1) where BC is the same matrix as in Eq. (6.2). This matrix can be block-diagonalised by an orthogonal transformation of the form ( 0 BC a1 0 (bc)t 0 0 b1 a1 0 0 BC 0 b1 (BC)T 0 ? 12N ,7212 L2N 7212^ - 7212 L2N , (8.2) The adjacency matrix after the transformation is: I a1 BC 0 V 0 BC b1 0 0 0 0 - a1 (bc) T 0 0 BC -b1 AC 0 0 AC (8.3) in which AC is the adjacency matrix of C2N (a, b | c, d) as shown in Eq. (6.1), and AC is the adjacency matrix of C2N(-a, -b | c, d), the ring with vertex weights of opposite sign. It follows that we can use the results of Section 6 to derive expressions for all relevant quantities. The eigenvalues for T4N(a, b | c, d) are therefore E E T(s) k± T(a) k± EC±, EC± for k = 1, 2, (8.4) where EC is the expression given in Eq. (6.10) for the eigenvalues of the cycle C2N (a, b | c, d), and EC refers to the eigenvalues of the cycle C2N (-a, -b | c, d) with reversed vertex T P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 401 weights. The eigenvectors for the treadmill can be placed in a vector cT, with entries cT(s) _ cpk± _ cT(s) _ cpk± _ 1 cP V2 cPk± cT(a) _ cpk± _ cT(a) _ — cpk± _ 1 cP V2 cPk± (8.5) where the superscripts (s), and (a) indicate the symmetric and antisymmetric eigenvectors, respectively. 8.2 Flipped treadmills, TfN (a, b | c, d) Flipped treadmills can be obtained by simply changing the sign of a pair of symmetrically positioned edges on the top and bottom rings. We shall take edges (1, 2N) and (I, 2N) to have weights -d, without loss of generality, since a series of orthogonal transformations can move the flipped pair of edges to any position around the rings. We use the same diagonalization methodology as for the standard treadmill, whence the adjacency matrix is ( al BC/ 0 \ 0 BCf 0 0 bl 0 0 0 -al Bcf (BC' )T 0 -bl AC 0 Ac (8.6) where the matrix Bcf is shown in Eq. (6.15), Acf is the adjacency matrix of CfN(a, b —f -f c, d) as shown in Eq. (6.14), and AC is the adjacency matrix of CfN(-a, —b | c,d), the ring with vertex weights of opposite sign. The eigenvalues for TfN (a, b | c,d) are thus ETf (s) _ Ec E i _ Ek Tf (a) k± Cf Ek± for k _ 1, 2, ,N, (8.7) where EC±, given in Eq. (6.20), is an eigenvalue of CfN(a, b | c, d), and EC± is an eigenvalue of CfN(-a, —b | c, d). The eigenfunction entries can be written as Tf (s) cpk± cTf (a) cpk± cTf(s) cpk± Tf (a) cpk± 1 ccf 72 cpk±, 1 nf (8.8) V2 cPk±, with 2N eigenvectors in the symmetric and the antisymmetric blocks (i.e. k _ 1, 2,..., N). 8.3 Möbius treadmills, TMN(a, b | c,d) The Möbius treadmill has the same involution symmetry as the other treadmills defined in Section 8. Using the treadmill block diagonalisation procedure, the adjacency matrix T 0 f f 402 Ars Math. Contemp. 13 (2017) 275-291 C 2N d 1 c 2 d 3 c 4 2N-2 d 2N-1 c a b a b b c 2N 1 c 2 d 3 c 4 2N-2 d 2N-1 c Figure 6: A Möbius treadmill, T4MN(a, b | c, d), with 4N vertices and alternating rung (a, b) and ring edge weights (c, d). a becomes / a1 (bc)t 0 \ 0 BC 61 0 0 0 0 —a1 BC 0 0 Cf B -61 AC 0 ACf (8.9) which has the structure of C2N(a, b | c, d) in the top block, and CfN(-a, —b | c, d) in the lower one. The eigenvalues for TjN (a, b | c, d) are thus E TM (s) k± Me k± E±, ETM(a) _ ECf _ Ek± for k _ 1, 2, (8.10) The eigenfunction entries can be written as cTM (s) cpk± Tm (a) cTM (s) °pk± cTM (a) — °pk± V2 ^ 1 JCf cpk± = cpk± = CPk±, with 2N eigenvectors in the symmetric and the antisymmetric blocks. 8.4 Flipped Möbius treadmills, T^f (a, b | c,d) The block diagonalisation procedure gives the adjacency matrix (8.11) a1 (bC )T 0 0 BCf T 61 0 0 0 0 —al BC 0 0 (bc)t —61 A Cf 0 AC , (8.12) which has the structure of (a, b | c, d) in the top block, and C2N (—a, —b | c, d) in the lower. The eigenfunction entries, in this case, are V2 ^ 1 , cCk±. (8.13) cTMf (s) cpk± TMf (s) °pk± rpMf / \ rpMf / \ I cT (a) _ cT (a) _ cpk± _ °pk± _ ^ cp V2 0 T 0 P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 403 9 Structural polynomials of alternating treadmills The characteristic polynomial, sT4N (E), can be written immediately because we have been able to split the secular equations into two annulene terms, C2N (a, b | c, d) and C2N(-a, —b | c, d), to give sT4N = s(C2N(a, b | c, d), E)s(C2N(—a, —b | c, d), E) = 4(cd)2N (Tn(x) — 1) (Tn(x) — 1), (9.1) with x and x defined in Eqs. (3.5) and (5.2). We can also write sT4N = 4(cd)2N (T2n(y) — 1) (T2n(y) — 1) (9.2) with y and y defined in Eqs. (3.10) and (5.4). The factoring of the secular problem for the treadmill allows the characteristic polynomials to be written as a product of the characteristic polynomials of C2N(a, b | c, d) and C2N(—a, —b | c, d). The other structural polynomials can be obtained using the same logic as in Section 5, = 1 (XC2N(a, b | c, d), E)s(C2N(—a, —b | c, d), E) + s(C2N(a, b | c, d), E)^(C2n(—a, — b | c, d), E)). (9.3) It follows that, for pairs of indices on the same ring, ^TpN2p+2q = (cd)2N— 1 {(E — a) [Un q 1 (x) + Uq_i(x)j [Tw(x) — 1] + (E + a) [Un-q-i(x) + Uq_i(x)] [Tn (x) — 1]} , jTpil,2p+2q+1 = (cd)2N-1 {(E — b) [Un q 1 (x) + Uq-1 (x)] [T* (x) — 1] + (E + a) [Un—q— 1 (x) + Uq— 1 (x)] [Tn (x) — 1]} , $?1,2p+2q+2 = (cd)2N —1 {[Un — q—1(x) + Uq(x)] [T* (x) — 1] + [Un—q—1(x) + Uq(x)] [Tn(x) — 1]} , (9.4) and for pairs of indices on different rings, •?Tp"2p+2q = (cd)2N —1 {(E — a) [Un—q—1(x) + Uq—1(x)] [Tn(x) — 1] — (E + b) [Un — q— 1(x) + Uq—1(x)] [Tn (x) — 1]} , jTp+1,2p+2q+1 = (cd)2N —1 {(E — b) [Un — q—1(x) + Uq—1(x)] [T* (x) — 1] — (E + b) [Un — q— 1(x) + Uq—1(x)] [Tn (x) — 1]} , jTp + 1,2p+2q+2 = (cd)2N —1 {[Un — q—1(x) + Uq (x)] [Tn (x) — 1] — [Un—q—1(x) + Uq(x)] [Tn(x) — 1]} , (9.5) where we observe the expected sign change arising from the antisymmetry. 9.1 Structural polynomials of TfN(a,b | c,d) The characteristic polynomial, st4n (E), can be written immediately as we have been able to split the secular equations into contributions from two flipped annulenes, CfN (a, b | c, d) 404 Ars Math. Contemp. 13 (2017) 275-291 and Cfv(-a, -b | c, d), to give st4n = s(C2fN(a, b | c, d), E)s(CfV(-a, -b | c, d), E) = 4(cd)2V (Tv(x) + 1) (Tv(x) + 1), (9.6) with x and x defined in Eqs. (3.5) and (5.2). We can also write a product form sT4fN = 4(cd)2N (T2N(y) + 1) (T2N(y) + 1) (9.7) with y and y defined in Eqs. (3.10) and (5.4). The j structural polynomials can be obtained using the same logic as in Section 5: 3tÂn = 1 (j(Cv(a, b | c, d), E)s(Cfv(-a, -b | c, d), E) + S(Gfv(a, b | c, d), E)j(Cfv(-a, -b | c, d), E)) , (9.8) so that, for indices on the same ring, JTpNN2p+2q = (cd)2v— 1 {(E - a) [Uv q 1 (x) - Uq_i(x)j [Tv(x) + 1] + (E + a) [Un-q-i(x) - Uq_i(x)] [Tv (x) + 1]} , JTf+1,2p + 2q+1 = (cd)2N-1 {(E - b) [Uv q 1 (x) - Uq-l(x)] [Tv (x) + 1] + (E + a) [Uv—q— 1 (x) - Uq— 1 (x)] [Tv (x) + 1]} , JTf+1,2p + 2q+2 = (cd)2N —1 {[Uv — q—1(x) - Uq(x)] [Tv (x) + 1] + [Uv — q—1(x) - Uq(x)] [Tv (x) + 1]} , (9.9) and, for indices on different rings, JTfN2p+2q = (cd)2N —1 {(E - a) [Uv—q—1(x) - Uq—1(x)] [Tv(x) + 1] - (E + b) [Uv — q—1(x) - Uq—1(x)] [Tv (x) + 1]} , JTfi1,2p + 2q+1 = (cd)2V —1 {(E - b) [Uv — q—1(x) - Uq—1(x)] [Tv (x) + 1] - (E + b) [Uv — q—1(x) - Uq—1(x)] [Tv (x) + 1]} , J&1M+2 = (cd)2V —1 {[Uv—q—1(x) - Uq(x)] [Tv(x) + 1] - [Uv—q—1(x) - Uq(x)] [Tv(x) + 1]} . (9.10) The differences between equations for plain (c.f. Eqs. (9.4) and (9.5)) and flipped treadmills occur because of the sign changes arising from edge flips. These affect all terms inside the square brackets because symmetry divides the adjacency matrix into two flipped rings Cfv. 9.2 Structural polynomials of TjN(a, b | c,d) The characteristic polynomial, s^ (E), can be written immediately as we have been able to split the secular equations into contributions from two annulenes, C2V(a, b | c, d) and P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 405 CfN(-a, —b | c, d), to give stMN = s(C2N(a, b | c, d), E)s(CfN(—a, —b | c, d), E) = 4(cd)2N (Tn(x) — 1) (Tn(x) + 1), (9.11) with x and x defined in Eqs. (3.5) and (5.2). We can also write = 4(cd)2N (T2n(y) — 1) (T2n(y) + 1), (9.12) with y and y defined in Eqs. (3.10) and (5.4). The j structural polynomials can be obtained using the same logic as in Section 5, T M 1 / , Jpf = ^ (j(C2N(a, b | c, d), EMCn(—a, —b | c, d), E) + s(C2N(a, b | c, d), E)j(CfN(—a, — b | c, d), E)) , (9.13) so that, for indices on the same ring, jT?2p+2q = (cd)2N— 1 {(E — a) [Un_q_1 (x) + Uq—ftx)] [Tn(x) + 1] + (E + a) [Un—q—1(x) — Uq—ftx)] [Tn (x) — 1]} , jS+1,2p+2q+1 = (cd)2N— 1 {(E — b) [Un+q 1 (x) + Uq 1 (x)] [Tn(x) + 1] + (E + a) [Un_q_1(x) — Uq—ftx)] [Tn (x) — 1]} , jS+1,2p+2q+2 = (cd)2N— 1 {[Un q 1 (x) + Uq(x)] [Tn(x) + 1] + [Un—q— 1 (x) — Uq(x)] [Tn(x) — 1]} , (9.14) whilst for indices on different rings, JTM2p+2q = (cd)2N—1 {(E — a) [Un—q—1(x) + Uq— ftx)] [Tn(x) + 1] — (E + b) [Un—q—1(x) — Uq—1(x)] [Tn (x) — 1]} , Jtm)2p+1,2p+2q+1 = (cd)2N—1 {(E — b) [Un—q—1(x) + Uq—1(x)] [Tn(x) + 1] — (E + b) [Un—q—1(x) — Uq—1(x)] [Tn (x) — 1]} , J§+1,2p+2q+2 = (cd)2N—1 {[Un—q—1(x) + Uq(x)] [Tn(x) + 1] — [Un—q—1(x) — Uq(x)] [Tn (x) — 1]} . (9.15) The sign changes in this case affect only one of the sets of terms inside square brackets, because symmetry divides the adjacency matrix into contributions from one ring C2N and one flipped ring CfN. 9.3 Structural polynomials of T^J (a, b | c, d) The characteristic polynomial, sTM (E), can be written immediately as we have been able to split the secular equations into contributions from annulenes, CfN (a, b | c, d) and C2N(—a, —b | c, d), to give T Mf f s4N = s(C2fN(a, b | c, d), E)s(C2N(—a, —b | c, d), E) = 4(cd)2N (Tn(x) + 1) (Tn(x) — 1), (9.16) 406 Ars Math. Contemp. 13 (2017) 275-291 with x and x defined in Eqs. (3.5) and (5.2). We can also write sTf = 4(cd)2N (T2n(y) + 1) (T2N(y) - 1), (9.17) with y and y defined in Eqs. (3.10) and (5.4). The j structural polynomials can be obtained using the same logic as in Section 5, TMf 1 / f JpT = i (j(C2fN(a, b | c, d), E)s(C2N(-a, -b | c, d), E) + S(Cn(a, b | c, d), E)j(C2N(-a, -b | c, d), E)) , (9.18) so that, for indices on the same ring, T Mf „.r , JTpN2p+2q = (cd)2N-1 {(E - a) [Un q 1 (x) - Uq-l(x)] [Tn(x) - 1] + (E + a) [Un-q-i(x) + Uq_i(x)] [Tn(x) + 1]} , T Mf „.r , JTp^i,2p+2q+i = (cd) — 1 {(E - b) [Un q i(x) - Uq i(x)] [Tn(x) - 1) + (E + a) [Un-q-i(x) + Uq_i(x)] (Tn(x) + 1]} , T Mf „,r JTp^i,2p+2q+2 = (cd) — 1 {[Un q i(x) - Uq(x)] [Tn(x) - 1] + [Un—q— i(x) + Uq(x)] [Tn(x) + 1]} , (9.19) and for indices on different rings, T Mf „.r , J2Tp^p+2q = (cd)2N —1 {(E - a) [Un—q—i(x) - Uq—i(x)] [Tn(x) - 1] - (E + b) [Un—q—i(x) + Uq—i(x)] [Tn(x) + 1]} , T Mf „.r , JTp^i,2p+2q+i = (cd) —1 {(E - b) [Un—q—i(x) - Uq—i(x)] [Tn(x) - 1) - (E + b) [Un—q—i(x) + Uq—i(x)] [Tn(x) + 1]} , T Mf „.r , jTP+iM+2 = (cd)2N —1 {[Un—q—i(x) - Uq(x)] [Tn(x) - 1] - [Un—q—i(x) + Uq(x)] [Tn(x) + 1]} . (9.20) The sign changes in this case affect only one of the sets of terms inside square brackets, because symmetry divides the adjacency matrix into a flipped ring CfN and one plain ring C2N. 10 Graphs derived from alternating ladders and treadmills A series of interesting graphs can be derived from our alternating ladders and treadmills by putting either a = 0, or b = 0. Some of the graphs that can be derived from ladders are shown in Fig. 7. Ladders with backbone chains with odd numbers of vertices lead to polyacenes with arms and legs, or to polyacenes themselves (Fig. 7(a) and 7(b)), by putting the first rung edge parameter or the second to zero. Even-vertex backbones give polyacenes with a single arm and leg, as shown in Fig. 7(c), whichever rung weight is set to zero. In the case of treadmills, it does not matter which rung weight is set to zero. In either case one obtains cyclic polyacenes. The appropriate formulae in Sections 4 and 8 for eigenvalues, eigenvectors and structural polynomials can be used in these cases. P. W. Fowler et al.: Spectra and structural polynomials of graphs of relevance to the theory ... 407 (a) (b) (c) Figure 7: Graphs derived from ladders by zeroing rung parameters to zero: (a) L14(0,1 | 1,1), with two 7-vertex backbone chains, a = 0; (b) L14(0,1 | 1,1), with two 7-vertex backbone chains, b = 0; (c) L12(1,0 | 1,1), with two 6-vertex backbone chains, b = 0. 11 Conclusions The algebraic development of structural polynomials reported here has been carried out in order to have exact results on which to base an elaboration of the theory of molecular conduction. The new formulae will allow us to treat n distortivity and its effect on ballistic conduction through conjugated molecular frameworks, as predicted within the source-sink potential (SSP) approach, a model that has a very direct connection to graph theory. In the simplest picture, a molecular device consists of a molecule attached to two semi-infinite wires. Such a device can be modelled qualitatively by replacing the system by a graph in which the electronic interactions are replaced by a series of edge weights. Furthermore, the infinite wires can be replaced by source and sink vertices, with complex vertex weights modified to reproduce the physics of a current of electrons down the wires. The formula for the transmission, T(E), of current of electrons through a device with energy E, can then be expressed in terms of four polynomials derived from the graph of the molecule. These polynomials can be chosen as the characteristic polynomial s(E) of the graph itself, jpp(E), jqq(E), and ,?pq(E) (c.f. Eqs. (1.1) and (1.2)), where vertices p and q of the molecule are attached to the source and sink vertices in the device. The formulae we have obtained for ladders and the various forms of treadmills show that the structural polynomials can be written in a simple manner. We have shown that the existence of an involution allows the characteristic polynomials to be written neatly as a product of the characteristic polynomials of certain 'half' graphs comprising vertex-weighted backbones. The remaining structural polynomials are also expressed in terms of half graphs, albeit in a slightly more complicated form. Representation of the various structural polynomials in this 'factorised' form has advantages for understanding the structure of the spectrum and has implications for the physics of the transmission as a function of energy. The different sign patterns of the structural polynomials exhibited in, for example, the varieties of treadmill (flipped, Möbius, etc.) will have a profound effect on transmission, T(E), as a function of the energy of the incoming electrons. 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