A note on the graph’s resolvent and the multifilar structure

Linear Algebra and its Applications 431 (2009) 1367–1379 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa A note on the graph’s resolvent and the multifilar structure聻 Vladimir Ejov a , Shmuel Friedland b , Giang T. Nguyen a,∗ a b School of Mathematics of Statistics, University of South Australia, Australia Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, United States A R T I C L E I N F O Article history: Received 26 March 2009 Accepted 5 May 2009 Available online 28 June 2009 Submitted by Gutman AMS classification: 05C45 05C75 11M36 A B S T R A C T We consider the recently discovered [4] threadlike structure of the plot representing d-regular graphs in the mean–variance coordinates of exponential sums of the graph spectra. In this note we demonstrate that this self-similar phenomenon is more ubiquitous by exhibiting it with the help of a different generating function, namely the mean and the variance of the resolvent of the adjacency matrix of the graph. We also discuss the location of non-Hamiltonian graphs within this geometric structure. © 2009 Elsevier Inc. All rights reserved. Keywords: Regular graph Spectrum Resolvent Self-similar Ihara–Selberg trace formula 1. Introduction Let G be a connected cubic graph G of order n, and A be the adjacency matrix of G. The spectrum [−3, 3]. The Estrada index of {λ1 , . . . , λn } of A of a cubic graph G is real and belongs to the segment  the graph is introduced in [9] (see also, for example, [3,1]) as EE (G) = 1n ni=1 eλi , and the generalized Estrada index is introduced in [7] as 聻 The authors gratefully acknowledge the Australian Research Council (ARC) Discovery Grants Nos. DP0666632 and DP0984470, and the ARC Linkage International Grants Nos. LX0560049 and LX0881972. ∗ Corresponding author . E-mail addresses: [email protected] (V. Ejov), [email protected] (S. Friedland), [email protected] (G.T. Nguyen). 0024-3795/$ - see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2009.05.019 1368 V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 EE (G, t ) = n 1 et λi . n i=1 (1.1) In [4], the following phenomenon was discovered. As the generalized Estrada index is the mean of the exponential sum of all eigenvalues of tA, the authors consider the variance of the exponential sum of all eigenvalues of tA as follows: 2 σEE (G,t ) = Letting t n n 1 n i =1 e2t λi − EE 2 (G, t ). (1.2) 2 ) across the set of all cubic graphs. The structures for = 13 , the authors plot (EE (G, 13 ), σEE (G, 1 ) = 14 and 16 appear as in Figs. 1.1 and 1.2. 3 From these plots, we can see that the mean–variance coordinates form thread-like clusters with 2 similar slopes of and distances between consecutive clusters. Moreover, the variance σEE (G,1/3) at the bottom of each segment is strictly increasing from left to right of the plot. The authors of [4] coined the term multifilars to refer to these thread-like clusters, each of which is called a filar. They make an important observation that the overall structure is self-similar. In particular, zooming in on each of these filars shows us similar but smaller sub-filars that are also made up of approximately straight and parallel segments, shifted gradually from left to right. We illustrate this by showing plots of two successive enlargements of the first filar from Figs. 1.1 and 1.2, in Figs. 1.3 and 1.4, respectively. Using a form of Ihara–Selberg trace formula derived in [10], the authors explain the filar memberships for each graph. In the overall clustering, all graphs belonging to each segment have the same number of triangles (cycles of length three) and these numbers strictly increase from the left most segment to the right most, starting from zero. In the first level of zooming-in, all graphs in a particular sub-segment have the same number of quadrangles (cycles of length four) while the number of triangles over all these sub-segments is fixed. This pattern repeats itself, with each higher level of zooming-in corresponding to a larger cycle size. It is natural to query whether the exponential function is the only generating function that exhibits such a phenomenon. In this paper, we consider another frequently used matrix function in the spectral theory of linear operators (see, for example, [12]), that is, the resolvent of tA for t ∈ (0, 13 ). It appears the phenomenon in [4] is reproduced in the mean–variance coordinates with different slopes of and distances between segments. We use a modification of the Ihara–Selberg trace formula [10] to justify 0.58 0.56 0.52 0.5 σ2 EE(G,1/3) 0.54 0.48 0.46 0.44 0.42 1.17 1.175 1.18 1.185 1.19 EE(G,1/3) Fig. 1.1. Mean–variance plot for 14-vertex cubic graphs. 1.195 1369 V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 0.58 0.56 0.52 0.5 σ2 EE(G,1/3) 0.54 0.48 0.46 0.44 0.42 1.165 1.17 1.175 1.18 1.185 1.19 1.195 1.2 EE(G,1/3) Fig. 1.2. Mean–variance plot for 16-vertex cubic graphs. 0.49 0.47 0.46 σ2 EE(G,1/3) 0.48 0.45 0.44 0.43 0.42 1.172 1.174 1.176 1.178 1.18 1.182 EE(G,1/3) Fig. 1.3. Zooming in, 1st level. the multifilar structure of the observed plots and to give estimate for the slopes of and distances between segments, which are consistent with numerical evidence. The further question: what other generating functions apart from the exponential sum of eigenvalues and the trace of the resolvent lead to a similar self-similar structure of regular graphs, is beyond the scope of this paper and remains an interesting open question. One of the famous graph theory problems is the Hamiltonian cycle problem, namely, given a graph, we have to determine whether there exists a simple cycle that goes through every vertex in the graph. Such a cycle is called a Hamiltonian cycle. If a graph possesses at least one Hamiltonian cycle, it is called a Hamiltonian graph, and a non-Hamiltonian graph otherwise. We observe that in the aforementioned self-replicating phenomenon, non-Hamiltonian graphs are separated in two groups. The first group contains easy non-Hamiltonian graphs that are located at the tops of (the most zoomed in) sub-filars. 1370 V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 0.465 0.46 σ2 EE(G,1/3) 0.455 0.45 0.445 0.44 0.435 0.43 1.173 1.174 1.175 1.176 1.177 1.178 EE(G,1/3) Fig. 1.4. Zooming in, 2nd level. We call a non-Hamiltonian graph easy if it contains one or more bridges. This is because these bridges can be identified in polynomial time [13]. We call other non-Hamiltonian graphs hard. The second group contains hard non-Hamiltonian graphs that are found at the bottom ends of (the most zoomed in) sub-filars. In these sub-filars, the Hamiltonian graphs are strictly in between these two groups of non-Hamiltonian graphs. 2. Preliminaries We briefly describe here a few definitions on geodesics that will be necessary for presenting the results. For an excellent introduction to graph theory and for more details on geodesics, the interested reader is referred to [2,10], respectively. An elementary homotopy is a transformation of a closed walk of the following form: (v1 , v2 , . . . , vi , . . . , vk−1 , vk , v1 ) → (v1 , v2 , . . . , vi , vj , vi , . . . , vk−1 , vk , v1 ), where vj is a neighbour of vi , and the arrow can also be pointing in the opposite direction: (v1 , v2 , . . . , vi , vj , vi , . . . , vk−1 , vk , v1 ) → (v1 , v2 , . . . , vi , . . . , vk−1 , vk , v1 ). If one closed walk can be obtained from another by a sequence of elementary homotopies (in either direction of the arrow), they are said to be homotopic. A homotopy class is a set of closed walks such that every pair of closed walks in the set are homotopic. In a homotopy class of closed walks, the shortest walk is called a closed geodesic. In other words, a closed geodesic is a closed walk with no cycles of length 2, that is, vi = / vi+2 for all i. As we are only concerned with closed geodesics in this paper, we will simply refer to them as geodesics. Also known as a short geodesic, a contractible is a geodesic of length 0, or equivalently, a single vertex. A homotopy class of closed walks containing a geodesic of length 0 is equivalent to a homotopy class of closed walks such that each member is either a single vertex or a union of two or more joint cycles of length 2. A long geodesic is a geodesic of length > 0, which, from now on, we will simply refer to as a geodesic. A geodesic of length 3, 4 or 5 is equivalent to a cycle of length 3, 4 or 5. On the other hand, a geodesic of length 6 or longer can be a union of joint cycles. Consider a geodesic g := (v1 , v2 , . . . , vl ) of length l. Another geodesic is said to be a k-multiple of g, denoted as g k , if it simply traces out g for k times: g k = ({v1 , v2 , . . . , vl }, {v1 , v2 , . . . , vl }, . . . , {v1 , v2 , . . . , vl }). A geodesic is said to be primitive if it is not a multiple of a shorter geodesic. V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 1371 3. Formulae for first and second moments = 1, . . . , N. ∈ 0, 13 in order to guarantee that the inverse of I − tA exists. For i = 1, . . . , N, it is clear that eigenvalues of (I − tA )−1 are (1 − t λi )−1 . For each adjacency matrix A, define the expected value function of (1 − t λi )−1 to be Let us denote of the adjacency matrix A of a given cubic graph by λi , for i  eigenvalues  We choose t µ(A, t ) := N 1  1 N i=1 1 − t λi = 1 N Tr[(I − tA )−1 ], (3.1) and the variance function to be 1 σ 2 (A, t ):= Tr[(I − tA )−2 ] − µ2 (A, t ) N = N 1  1 N i=1 (1 − t λi )2 For our experiments, we choose t = − µ2 (A, t ). 1 9  (3.2)  ∈ 0, 31 . Fig. 3.1 shows the plot of (µ(A ), σ 2 (A )) across all 509 cubic graphs of size 14, which exhibits a self-similar structure. In order to explain why pairs of coordinates of certain graphs belong to particular filars, we start by establishing alternative formulae for µ(A, t ) and σ 2 (A, t ). Let pℓ be the number of all walks of length ℓ in the graph G, for ℓ  0. It is well known (see, for example, [14]) that [A ℓ ]ii is the number of closed walks of length ℓ starting at i. Hence, 1   µ(A, t )= Tr (I − tA )−1 = = N 1  N n + tp1 1 N  Tr I + tA  + t 2 A2 + · · · + t 2 p2 + t 3 p3 + · · · = ∞ 1  N i =0  t i pi . (3.3) Note that, the infinite sum in (3.3) is a particular case of the generalized matrix functions introduced in [8] and it resembles the trace of the matrix exponential of tA 0.056 0.054 σ2(A,1/9) 0.052 0.05 0.048 0.046 0.044 0.042 1.0395 1.04 1.0405 1.041 1.0415 1.042 1.0425 1.043 1.0435 1.044 µ(A,1/9) Fig. 3.1. Mean–variance (trace of resolvent) plot for 14-vertex cubic graphs. 1372 V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 Tr[etA ] ∞ i  t = i! i =0 pi , Recall that a geodesic is the shortest walk of its homotopy class, and a geodesic is a short geodesic if it has length 0 and a long geodesic otherwise. Let C1 (t ) and C2 (t ) be the contributions to (3.3) from homotopy classes of short geodesics and long geodesics, respectively. Then, 1 µ(A, t ) = (C1 (t ) + C2 (t )) . N (3.4) By Eqs. (19) and (20) in [10], C1 (t ) =N 3 √ 1 − 8t 2 2(1 − −1 (3.5) . 9t 2 ) By Eqs. (26) and (27) in [10], C2 (t ) = Λ(γ )  √ γ ∈G 1 − 8t 2  √ 1− 1 − 8t 2 |γ | 4t , (3.6) where γ is a long geodesic in G, |γ | is its length and Λ(γ ) geodesic γ ′ . If γ is primitive itself, then Λ(γ ) = |γ |. √ Define Θ (t ) := 1− √ 1 − 8t 2 3 µ(A, t ) = 1−8t 2 . 4t 2 (1 − = |γ ′ | if γ is a multiple of a primitive Substituting (3.5) and (3.6) into (3.4), we have −1 9t 2 ) + 1  N γ ∈G Λ(γ ) √ 1 − 8t 2 Θ |γ | (t ). (3.7) Recall that a geodesic is said to be a k-multiple of g, denoted as g k , if it k times traces out g: g k = ({v1 , v2 , . . . , vl }, {v1 , v2 , . . . , vl }, . . . , {v1 , v2 , . . . , vl }). Consequently, if we consider some geodesic γ that is a k-multiple of a primitive geodesic γ ′ , then |γ | = k|γ ′ |. The set of all long geodesics in a graph G can be partitioned into sets of primitive geodesics and their k-multiples, for k = 1, . . . , N. Therefore, by denoting primitive long geodesics by ζ , we transform Eq. (3.7) to: µ(A, t )= = As t √ 3 1 − 8t 2 2 (1 − √ 3 −1 9t 2 ) + ∞ 1  N ζ ∈ G k =1 √ |ζ | 1 − 8t 2 Θ k|ζ | (t ) ∞  −1 |ζ | 1  √ Θ k|ζ | (t ). + 2(1 − 9t 2 ) N ζ ∈G 1 − 8t 2 k=1 1 − 8t 2 ∈ (0, 13 ), Θ (t ) ∈ (0, 1). Hence, √ 3 1 − 8t 2 − 1 + µ(A, t )= 2(1 − 9t 2 ) √ 3 1 − 8t 2 − 1 = + 2(1 − 9t 2 ) 1  √ 1  √ N ζ ∈G N ζ ∈G |ζ | 1 − 8t 2 |ζ | 1 − 8t 2 Θ |ζ | (t ) 1 − Θ |ζ | (t ) (1 − √ 1 − 8t 2 )|ζ | (4t )|ζ | − (1 − √ 1 − 8t 2 )|ζ | . (3.8) Let ℓ be the length of a primitive long geodesic, ℓ  3 and {m3 , m4 , m5 , . . .} be the length spectrum of the graph where mℓ is the number of non-oriented primitive long geodesics of length ℓ. We can rewrite (3.8) as µ(A, t ) = √ 3 √ ∞ ℓmℓ (1 − 1 − 8t 2 )ℓ 2  −1 √ √ + 2(1 − 9t 2 ) N ℓ=3 1 − 8t 2 (4t )ℓ − (1 − 1 − 8t 2 )ℓ 1 − 8t 2 1373 V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 = H (t ) + where H (t ) = 3 N ℓ=3 √ 1 − 8t 2 mℓ Fℓ (t ), −1 2(1 − 9t 2 ) and =√ Fℓ (t ) ∞ 2  By Eq. (3.2), , (3.10) (1 − ℓ 1 − 8t 2 (3.9) √ 1 − 8t 2 )ℓ (4t )ℓ − (1 − √ (3.11) . 1 − 8t 2 )ℓ 1 σ 2 (A, t )= Tr[(I − tA )−1 (I − tA )−1 ] − µ2 (A, t ) N 1 = Tr[I + 2tA + 3t 2 A 2 + 4t 3 A 3 + · · ·] − µ2 (A, t ) N 1 d [tI + t 2 A + t 3 A 2 + t 4 A 3 + · · ·] N dt  1 d  = tTr[(I − tA )−1 ] − µ2 (A, t ) N dt d = µ(A, t ) + t µ(A, t ) − µ2 (A, t ). dt = Tr − µ2 (A, t ) (3.12) By Eqs. (3.9) and (3.12), we have 2 ∞ 2  σ (A, t )= H (t ) + d 2 = H (t ) + tH (t ) − H (t ) + − 4 N2 where = −3 and Fℓ′ (t ) = ⎣H (t ) + m ℓ F ℓ (t ) + t N ℓ=3 dt ⎤2 ⎡ ∞ 2  2 ⎣ mℓ Fℓ (t )⎦ − H (t ) − N ℓ=3 ′ H ′ (t ) ⎡ ℓ(1 − √ 1 9  ⎣ ∞  ℓ=3 ⎤2 mℓ Fℓ (t )⎦ − N ℓ=3 4 N N ℓ=3 m ℓ F ℓ (t )⎦ mℓ H (t )Fℓ (t ) ⎛ m ℓ F ℓ (t ) + t ⎝ H (t ) ∞  ⎤ ∞ 2  N ℓ=3 ⎞ mℓ Fℓ′ (t )⎠ mℓ Fℓ (t ), (3.13) ℓ=3  √ −5 + 36 t 2 + 3 1 − 8 t 2 √ ,  2 1 − 8 t 2 −1 + 9 t 2 1 − 8t 2 )ℓ (8(4t )ℓ t 2 − 8(1 − In particular, for t µ A, t ⎡ ∞ 2  N ℓ=3 ∞ 4  − ∞ 2  √ 1 − 8t 2 )ℓ t 2 − 8(4t )ℓ t 2 t (−1 + √ 1 − 8t 2 ) √ √ 1 − 8t 2 + 8ℓ(4t )ℓ t 2 + 8(1 − 1 − 8t 2 (−1 + 8t 2 )(−(4t )ℓ + (1 − √ √ 1 − 8t 2 )ℓ √ 1 − 8t 2 )ℓ )2 ) √ 1 − 8t 2 − (4t )ℓ ℓ + ℓ 1 − 8t 2 (4t )ℓ ) . = 19 , Eqs. (3.9) and (3.13) simplify to, respectively, ≈ 1.0395 + ∞ 2  N ℓ=3 mℓ Fℓ 1 9 , and (3.14) 1374 V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 σ 2 A, 1 ≈ 0.0433 + 9 − 4 N2 ⎡ ⎣ ∞  ∞ 2  N ℓ=3 mℓ Fℓ ℓ=3 1 mℓ Fℓ 1 9 ⎤2 ⎦ + 9 − 1 9 ⎛ ⎝ ∞ 2  N ℓ=3 ∞ 4.1580  N   ⎞ ⎠ mℓ Fℓ′ (t ) t = 19 1 mℓ Fℓ 9 ℓ=3 , (3.15) where √  √ ℓ ℓ 73 1 − 19 73 =    √  , 9 73 4 ℓ − 1 − 1 73 ℓ 9 9 9 1 Fℓ and   Fℓ′ (t ) t = 19 = 81 5329 ℓ(9 − √ 73)ℓ (−72(4)ℓ √ √ √ √ √ + 72 73(9 − 73)ℓ − 584 (9 − 73)ℓ − 59134ℓ ℓ + 657 73ℓ4ℓ ) √ √ . ℓ ℓ 2 (−9 + 73)(−4 + (9 − 73) ) 73 + 584(4)ℓ Note that the details of these derivations can easily be verified using MATLAB 7.5.0 or MAPLE 9.5. 4. Rates of change and dominant terms Note that the following analysis of the rates of change, dominant terms, slopes of and distances between filars is similar to arguments presented in [4], in which the self-similar multifilar phenomenon was first discovered. For various fixed values of t ∈ (0, 13 ), our experiments show that both function Fℓ (t ) and its par  tial derivative Fℓ′ (t ) are rapidly decreasing as ℓ grows. In fact, we observe that Fℓ 91  C1 10−ℓ and [Fℓ′ (t )]t = 1  C2 10−ℓ for some positive constants C1 and C2 . It is reasonable to assume that on the other 9 hand, mℓ does not grow as fast as C10ℓ for some positive constant   C as ℓ increases. 1 9 As a result, the contribution of the quadratic terms of Fℓ in (3.15) is insignificant and ℓ=3 is the dominant term in the infinite sums in (3.14) and (3.15). Recall our observation that each filar, where each graph has m3 triangles, is made up of sub-filars. Each of these sub-flars consists of graphs that have exactly m4 rectangles, with m4 = 0 for the left-most sub-filar and m4 increases by 1 from one sub-filar to the next sub-filar to the right. Consequently, the lower endpoint of each filar is most likely to contain a graph that has m3 triangles and zero rectangles. Therefore, from (3.14) and (3.15), with t = 91 , we can approximate the coordinates for the lower end point of each filar with graphs possessing m3 triangles by: 2 1 m 3 F3 , and N 9 1 2 1 + 0.0433 + m3 F3 N 9 9 µ(m3 ) = 1.0395 + σ 2 (m3 ) = (4.1) 2 N  m3 F3′ (t )  t = 91 − 4.1580 N m 3 F3 1 9 . (4.2) Consequently, let k ∈ [0, ∞), then the line (see Fig. 4.1) that goes through the lower end points of filars can be approximated by the parametric line (x(k), y(k)), where 1 x (k ) = 1.0395 + kF3 y (k ) = 0.0433 + k F3 (t ) + 9 ≈ 1.0395 + 0.0047k, 1 ′  F3 (t ) t = 91 9 − 2.0790F3 (4.3) 1 9 ≈ 0.0433 + 0.0103k. (4.4) The slope of the line parametrically described by (4.3) and (4.4) (and represented by the black line in Fig. 4.1) is 0.0103/0.0047 = 2.1901, which is close to the experimental value of 2.04 for cubic graphs of 14 vertices. 1375 V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 0.056 0.054 2 σ (A,1/9) 0.052 0.05 0.048 0.046 0.044 0.042 1.0395 1.04 1.0405 1.041 1.0415 1.042 1.0425 1.043 1.0435 1.044 1.0445 µ(A,1/9) Fig. 4.1. Mean–variance (trace of resolvent) plot for 14-vertex cubic graphs. 0.056 0.054 0.05 2 σ (A,1/9) 0.052 0.048 0.046 0.044 0.042 1.039 1.04 1.041 1.042 1.043 1.044 1.045 1.046 µ(A,1/9) Fig. 4.2. Mean–variance (trace of resolvent) plot for 14-vertex cubic graphs. The observation that ℓ = 3 is is the dominant term in the infinite sums in (3.14) and (3.15) also explains why in the self-similar structure in Fig. 3.1 if two cubic graphs have the same number of triangles, then they belong to the same filar. Similarly, we can explain the membership of sub-filars of various levels. Each filar, where all graphs have m3 triangles, can be approximated (see Fig. 4.2) by a line parametrically defined as follows: x(s)= 1.0395 + = 1.0395 + 2 N 2 N m 3 F3 1 9 0.0047m3 + sF4 1 9 + 0.0007s, (4.5) 1376 V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 2 y(s)= 0.0433 + N 1 + s F4 = 0.0433 + 9 2 N m 3 F3 1 + 9 1 ′  F 3 (t ) t = 91 9 1 ′  F4 (t ) t = 91 9 + 0.0103m3 1 − 2.0790F3 − 2.0790F4 9 1 9 + 0.0023s. The slope of each filar is approximately:       F4 91 + 91 F4′ (t ) t = 1 − 2.0790F4 19  9 F4 19 (4.6) ≈ 3.2448, which is independent of the graph size N. Consider two consecutive filars, consisting of graphs con(1) (2) taining exactly m3 and m3 parametrically defined by: 2 x̄(s1 ) = 1.0395 + N ȳ(s1 ) = 0.0433 + N 2 triangles respectively. Then the line approximating the first filar is (1) + 0.0007s1 , (1) + 0.0023s1 , 0.0047m3 0.0103m3 and the line approximating the second filar is parametrically defined by: 2 x̂(s2 ) = 1.0395 + N ŷ(s2 ) = 0.0433 + N 2 (2) + 0.0007s2 , (2) + 0.0023s2 , 0.0047m3 0.0103m3 In order to find out the horizontal distance between two filars approximated by the parametric lines (x̄(s1 ), ȳ(s2 )) and (x̂(s1 ), ŷ(s2 )), firstly, we need to find out s1 and s2 such that ȳ(s1 ) = ŷ(s2 ): 0.0433 + 2 N (1) 0.0103m3 + 0.0023s1 = 0.0433 + (2) s1 = s2 + 0.0206 m3 2 N (2) 0.0103m3 + 0.0023s2 (1) − m3 . N Then the horizontal distance between the two aforementioned filars is: ⎛ ⎞ (2) (1) m 3 − m3 ⎠ x̂(s2 ) − x̄ ⎝s2 + 0.0206 N = 2 N (2) 0.0047m3 (2) = 0.0094 m3 + 0.0007s2 − (1) − m3 N 2 N (1) 0.0047m3 ⎛ − 0.0007 ⎝s2 (2) + 0.0206 m3 (1) ⎞ − m3 ⎠ N . Hence, the horizontal distance between two consecutive filars decreases as the graph size N increases, so the filars are closer to each other as the graphs get larger. V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 1377 5. Self-replicating structure and Hamiltonicity Due to the interest in the Hamiltonian cycle problem, we reconstruct the variance-versus-mean plot in Fig. between Hamiltonian and non-Hamiltonian graphs. Each pair of coordinates  3.1,  butdistinguish  (µA 1 9 , σA2 1 9 ) of a 14-vertex cubic graph is a dot on this reconstructed plot (see Fig. 5.1) if the graph is Hamiltonian, and a cross if the graph is non-Hamiltonian. Two graphs are cospectral if they share the same spectrum, and are non-cospectral otherwise. There are no cospectral cubic graphs with fewer than 14 vertices, and there are at least three pairs of cospectral cubic graphs on 14 vertices [11]. Therefore, there are three pairs of graphs of which the coordinates in the above plot are the same. However, as these three pairs are all Hamiltonian graphs, their cospectral property does not affect our observation on the self-similar multifilar structure and Hamiltonicity. 0.056 0.054 σ2(A,1/9) 0.052 0.05 0.048 0.046 0.044 0.042 1.0395 1.04 1.0405 1.041 1.0415 1.042 1.0425 1.043 1.0435 1.044 µ(A,1/9) Fig. 5.1. Mean–variance plot for 14-vertex cubic graphs and Hamiltonicity. 0.047 0.0465 0.0455 2 σ (A,1/9) 0.046 0.045 0.0445 0.044 1.0398 1.04 1.0402 1.0404 µ(A,1/9) Fig. 5.2. Mean–variance plot – zooming in. 1.0406 1.0408 1378 V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 0.056 0.054 2 σ (A,1/9) 0.052 0.05 0.048 0.046 0.044 1.0395 1.04 1.0405 1.041 1.0415 1.042 1.0425 1.043 1.0435 1.044 µ(A,1/9) Fig. 5.3. Mean–variance plot for 10-vertex cubic graphs and Hamiltonicity. Fig. 5.4. The only cubic bridge graph of size 10. Glancing at the plot in Fig. 5.1, it is easy to think that while the majority of non-Hamiltonian graphs are located in the top and bottom parts of filars, some of them are mixed among dots representing Hamiltonian graphs. However, if we zoom in on the innermost1 sub-filars, we will find that all nonHamiltonian graphs are strictly at the top and the bottom of these sub-filars, and there is no mixing between Hamiltonian and non-Hamiltonian graphs. While it is not practical to show all plots which zoom in on the innermost sub-filars, we present here one of these zooming-in plots (see Fig. 5.2) to illustrate our observation. All crosses that can be seen clearly in this plot are either at the top or the bottom of their sub-filars. We have not yet been able to explain why the absence of Hamiltonian cycles make non-Hamiltonian graphs gather around the top and the bottom of their innermost sub-filars. However, we have experimentally found the answer as to which non-Hamiltonian graphs are at the higher end while the rest are at the lower. It is worth noting that the following work is in similar spirit to [5,6]. Briefly, non-Hamiltonian graphs that are located at the top of their sub-filars are bridge graphs. A bridge graph is a graph that contains at least one bridge, that is, an edge the removal of which disconnects the graph. Section 57 in [13] contains a very natural theorem which states that all bridge graphs are non-Hamiltonian. These graphs can be identified in polynomial time; hence, we refer to them as easy non-Hamiltonian graphs. A non-Hamiltonian graph that is not an easy non-Hamiltonian graph is a hard non-Hamiltonian graph. The latter group are found to be at the bottom of their sub-filars. These hard non-Hamiltonian graphs constitute the underlying difficulty of the NP-hard complexity    to the Hamiltonian cycle problem. For example, Fig. 5.3 shows the plot of coordinates (µA 1 9 , σA2 1 9 ) for 10-vertex (connected) cubic graphs, of which there are 19, including 17 Hamiltonian ones. 1 When we can no longer zoom in on a filar to obtain a similar structure made up of smaller filars, we are at the innermost sub-filars. V. Ejov et al. / Linear Algebra and its Applications 431 (2009) 1367–1379 1379 Fig. 5.5. Petersen graph. One non-Hamiltonian graph is a bridge graph (see Fig. 5.4), represented by the cross at the top right of the plot. The other non-Hamiltonian graph is the well-known Petersen graph (see Fig. 5.5), which is not a bridge graph and is represented by the cross at the bottom left of the same plot. It would be interesting to obtain a theoretical justification of this observation. Acknowledgments The authors would like to thank J.A. Filar and P. Zograf for numerous useful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] J.A. de la Peña, I. Gutman, R. Rada, Estimating the Estrada index, Linear Algebra Appl. 427 (2007) 70–76. F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969. I. Gutman, Ante Graovac, Estrada index of cycles and paths, Chem. Phys. Lett. 436 (2007) 294–296. V. Ejov, J.A. Filar, S.K. Lucas, P. Zograf, Clustering of spectra and fractals of regular graphs, J. Math. Anal. Appl. 333 (2007) 236–246. E. Spectral Estrada, Spectral scaling and good expansion properties in complex networks, Europhys. Lett. 73 (2006) 649–655. E. Estrada, Network robustness. The interplay of expansibility and degree distribution, Eur. Phys. J. B Condens. Matter Phys. 52 (2006) 563–574. E. Estrada, N. Hatano, Statistical-mechanical approach to subgraph centrality in complex networks, Chem. Phys. Lett. 439 (2007) 247–251. E. Estrada, D.J. Higham, Networks properties revealed through matrix functions, University of Strathclyde Mathematics Research Report 17, 2008. E. Estrada, J.A. Rodríguez-Velázquez, Subgraph centrality in complex networks, Phys. Rev. E 71 (2005) 056103. P. Mnev, Discrete path integral approach to the Selberg trace formula for regular graphs, Comm. Math. Phys. 274 (2006) 233–241. J.A. Filar, A. Gupta, S.K. Lucas, Connected co-spectral graphs are not necessarily both Hamiltonian, Austral. Math. Soc. Gaz. 32 (3) (2005) 193. V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Series: Operator Theory: Advances and Applications, 139 (2007), Birkhäuser. A. Sainte-Lague, Les reseaux (ou graphes), Mem. Sci. Math. 18 (1926.) E. van Dam,W. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003) 241–272.