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A Few Examples and Counterexamples in Spectral Graph Theory Discussiones Mathematicae Graph Theory 40 (2020) 637{662 doi:10.7151/dmgt.2275 A FEW EXAMPLES AND COUNTEREXAMPLES IN SPECTRAL GRAPH THEORY Dragan Stevanovic´ Mathematical Institute Serbian Academy of Sciences and Arts, Belgrade, Serbia e-mail: [email protected] Nikola Milosavljevic´ University of Niˇs Faculty of Sciences and Mathematics, Niˇs,Serbia e-mail: [email protected] and Damir Vukiceviˇ c´ University of Split Faculty of Sciences and Mathematics, Split, Croatia e-mail: [email protected] Dedicated to the memory of Professor Slobodan Simi´c. Abstract We present a small collection of examples and counterexamples for se- lected problems, mostly in spectral graph theory, that have occupied our minds over a number of years without being completely resolved. Keywords: communicability distance, spectral radius, integral graph, sec- ond Zagreb index, Wiener index, estrada index, almost cospectral graphs, NEPS of graphs. 2010 Mathematics Subject Classification: 05C50. 638 D. Stevanovic,´ N. Milosavljevic´ and D. Vukiceviˇ c´ 1. Introduction In the forthcoming sections we review selected problems, mostly in spectral graph theory, that were either posed in literature or that we came across in our research. Their common property is that they are all only partially resolved, despite our best efforts. Hopefully, readers of this special issue will find them interesting and will help to solve them completely. To avoid repetition in the following sections, we give here some common definitions. All graphs considered are simple and connected. The vertex and edge sets of a simple graph G are denoted by V (G) and E(G), respectively, while the adjacency matrix of G is denoted by A(G). If the graph G is known from the context, we will drop it as the argument and write just V , E, A, etc. The degree of a vertex v 2 V is denoted by dv, with δ and ∆ denoting the minimum and the maximum vertex degree in G, respectively. For a graph G with n vertices, we denote by λ1 ≥ · · · ≥ λn the eigenvalues of A, and with x1; : : : ; xn corresponding eigenvectors which form an orthonormal basis. The largest eigenvalue λ1 is also the spectral radius of A. We denote by Kn, Pn and Sn the complete graph, the path and the star on n vertices, respectively. The broom graph Br;s is obtained by identifying an endvertex of the path Pr with the center of the star Ss+1, so that Br;s has r + s vertices. 2. Communicability Distance A P Ak Let us deal with counterexamples first. With e = k≥0 k! , Estrada [13] defined the communicability distance between vertices u and v of a graph G as q A A A ξuv = (e )uu + (e )vv − 2(e )uv and further introduced the communicability distance sum Υ, an analogue of the Wiener index, as 1 X Υ(G) = ξ : 2 uv u6=v Estrada then posed the following conjectures. ∼ Conjecture 1 [13]. If G =6 Kn is a simple connected graph on n vertices, then Υ(Kn) < Υ(G). Conjecture 2 [13]. If T is a tree on n vertices, then Υ(Sn) ≤ Υ(T ) ≤ Υ(Pn). The lollipop graph Lr;s is obtained by identifying a vertex of the complete graph Kr and an endvertex of the path Ps+1, so that the resulting graph has r+s vertices. A Few Examples and Counterexamples in ... 639 Conjecture 3 [13]. If G is a simple connected graph on n > 5 vertices, then Υ(G) ≤ Υ(Ln−2;2). We will disprove Conjecture 2 by showing that Υ(Pn) < Υ(Sn) holds for all sufficiently large n. To show this, we first need to represent Υ(G) more directly in terms of the eigenvalues and the eigenvectors of G. Let Λ = diag(λ1; : : : ; λn) T and Q = [x1 ··· xn], so that A = QΛQ is a spectral decomposition of A. Since eA = QeΛQT we have n A X λ (1) e uv = xj;uxj;vej j=1 for all u; v 2 V , so that 1 X 1 X q Υ(G) = ξ = (eA) + (eA) − 2(eA) 2 u;v 2 uu vv uv u6=v u6=v v n 1 X uX (2) = u x2 + x2 − 2x x eλj 2 t j;u j;v j;u j;v u6=v j=1 v n 1 X uX = u (x − x )2eλj : 2 t j;u j;v u6=v j=1 This representation enables us to get bounds on Υ(G) in terms of λ1. Theorem 4. If G is a simple connected graph on n vertices, then p n(n − 1) 2n λ1 (3) Υ(G) ≤ e 2 : 2 Proof. Since the eigenvectors x1; : : : ; xn are normalized, for each u; v 2 V we 2 2 have 2jxj;uxj;vj ≤ xj;u + xj;v ≤ 1, so that 2 2 2 xj;u − xj;v ≤ xj;u + xj;v + 2jxj;uxj;vj ≤ 1 + 1 = 2: Then v v p u n u n 1 X uX n(n − 1)uX n(n − 1) 2n λ1 Υ(G) ≤ 2eλj ≤ 2eλ1 = e 2 ; 2 t 2 t 2 u6=v j=1 j=1 where in the second inequality above we used eλj ≤ eλ1 for j = 1; : : : ; n. 640 D. Stevanovic,´ N. Milosavljevic´ and D. Vukiceviˇ c´ Theorem 5. If G is a simple connected graph on n vertices, then 0 1 1 1 λ1 (4) Υ(G) ≥ @ − p A e 2 : q δ n n − 1 + ∆ If G is not regular, then further 1 λ1 (5) Υ(G) ≥ p e 2 : 2n2 n Proof. By dropping nonnegative summands for j = 2; : : : ; n in the expression (3) for Υ(G) we get v n 1 X uX Υ(G) = u (x − x )2eλj 2 t j;u j;v u6=v j=1 λ1 1 X q e 2 X ≥ (x − x )2eλ1 = jx − x j: 2 1;u 1;v 2 1;u 1;v u6=v u6=v If x1;min = minu2V x1;u and x1;max = maxu2V x1;u denote the minimum and the maximum principal eigenvector component of G, respectively, then we can drop further nonnegative summands from the above inequality to obtain λ1 (6) Υ(G) ≥ (x1;max − x1;min)e 2 : Pn 2 2 p1 Since 1 = j=1 x1;j ≥ nx1;min, we have x1;min ≤ n . Cioaba and Gregory [4, Lemma 3.3] showed that 1 (7) x1;max ≥ ; q δ(G) n − 1 + ∆(G) with a stronger bound if G is not regular 1 1 (8) x1;max > ≥ : q 1 q 1 n − ∆(G) n − n−1 p1 Combining (6), x1;min ≤ n and (7), we directly obtain (4). If G is not regular, p1 then combining x1;min ≤ n and (8) yields A Few Examples and Counterexamples in ... 641 1 1 1 x1;max − x1;min ≥ − p = p p q 1 n p 2 2 2 n − n−1 n(n −n − 1) n − n + n −n−1 1 1 p ≥ p p p = 2 ; n · n2 n2 + n2 2n n which in combination with (6) yields (5). Now we can disprove Conjecture 2 for all sufficiently large n. Theorem 6. There exists n0 2 N such that Υ(Pn) < Υ(Sn) for all n ≥ n0. πj Proof. The eigenvalues of Pn are equal to 2 cos n+1 for j = 1; : : : ; n, so that λ1(Pn) < 2 and the upper bound (3) gives p n(n − 1) 2n Υ(P ) ≤ e: n 2 p On the other hand, the largest eigenvalue of Sn is n − 1 and the lower bound (5), since Sn is not regular for n ≥ 3, yields p 1 n−1 Υ(S ) ≥ p e 2 : n 2n2 n Since p n−1 p n−1 1p 2 −1 2n2 n e e 2 lim p = lim p = 1; n!1 n(n−1) 2n n!1 n4(n − 1) 2 2 e there exists n0 such that Υ(Sn) > Υ(Pn) for all n ≥ n0. Numerical results show that the smallest n for which Υ(Pn) < Υ(Sn) is n = 43. However, the path ceases to have the largest value of Υ among trees on a much smaller number of vertices. The double broom graph DBr;s;t is obtained by identifying one endvertex of Pr with the center of the star Ss+1 and the other endvertex of Pr with the center of the star St+1, so that DBr;s;t has r + s + t vertices. Then the three largest Υ values among trees on 15 vertices are Υ(P15) ≈ 199:60736; Υ(B13;2) ≈ 199:62532; Υ(DB11;2;2) ≈ 199:64285: As both paths and stars are special instances of brooms, our opinion is that it may be worthwhile to study further the behaviour of Υ(Br;s) and Υ(DBr;s;t), although that could not do much to save Conjecture 2 anyway. On the other hand, we could not find any counterexample for Conjectures 1 and 3. Conjecture 1 makes sense, as Kn is a regular graph with the all-one vector 642 D. Stevanovic,´ N. Milosavljevic´ and D. Vukiceviˇ c´ as the principal eigenvector, so that the summand corresponding to en−1 vanishes from (3), leaving only the summands corresponding to e−1 which make the value of Υ(Kn) smaller than Υ for many graphs whose eigenvalues are not bounded by a constant. The appearance of Ln−2;2 as the extremal graph in Conjecture 3 is somewhat unusual, despite a reasonable explanation provided by Estrada in [13]. As Kn is a special case of a lollipop as well, it would be worthwhile to study the behaviour of Υ(Lr;s) to confirm Conjectures 1 and 3 among lollipops at least.
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