On Some Aspects of the Generalized Petersen Graph

Let $p \ge 3$ be a positive integer and let $k \in {1, 2, ..., p-1} \ \lfloor p/2 \rfloor$. The generalized Petersen graph GP(p,k) has its vertex and edge set as $V(GP(p, k)) = \{u_i : i \in Zp\} \cup \{u_i^\prime : i \in Z_p\}$ and $E(GP(p, k)) = \{u_i u_{i+1} : i \in Z_p\} \cup \{u_i^\prime u_{i+k}^\prime \in Z_p\} \cup \{u_iu_i^\prime : i \in Z_p\}$. In this paper we probe its spectrum and determine the Estrada index, Laplacian Estrada index, signless Laplacian Estrada index, normalized Laplacian Estrada index, and energy of a graph. While obtaining some interesting results, we also provide relevant background and problems

The quotients between the (revised) Szeged index and Wiener index of graphs

Let $Sz(G),Sz^*(G)$ and $W(G)$ be the Szeged index, revised Szeged index and Wiener index of a graph $G.$ In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order $n\geqslant 10$ are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on $Sz(G)/W(G)$ is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least one cycle.Comment: 25 pages, 5 figure

Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP

The Traveling Salesman Problem asks to find a minimum-weight Hamiltonian cycle in an edge-weighted complete graph. Local search is a widely-employed strategy for finding good solutions to TSP. A popular neighborhood operator for local search is k-opt, which turns a Hamiltonian cycle C into a new Hamiltonian cycle C\u27 by replacing k edges. We analyze the problem of determining whether the weight of a given cycle can be decreased by a k-opt move. Earlier work has shown that (i) assuming the Exponential Time Hypothesis, there is no algorithm that can detect whether or not a given Hamiltonian cycle C in an n-vertex input can be improved by a k-opt move in time f(k) n^o(k / log k) for any function f, while (ii) it is possible to improve on the brute-force running time of O(n^k) and save linear factors in the exponent. Modern TSP heuristics are very successful at identifying the most promising edges to be used in k-opt moves, and experiments show that very good global solutions can already be reached using only the top-O(1) most promising edges incident to each vertex. This leads to the following question: can improving k-opt moves be found efficiently in graphs of bounded degree? We answer this question in various regimes, presenting new algorithms and conditional lower bounds. We show that the aforementioned ETH lower bound also holds for graphs of maximum degree three, but that in bounded-degree graphs the best improving k-move can be found in time O(n^((23/135+epsilon_k)k)), where lim_{k -> infty} epsilon_k = 0. This improves upon the best-known bounds for general graphs. Due to its practical importance, we devote special attention to the range of k in which improving k-moves in bounded-degree graphs can be found in quasi-linear time. For k <= 7, we give quasi-linear time algorithms for general weights. For k=8 we obtain a quasi-linear time algorithm when the weights are bounded by O(polylog n). On the other hand, based on established fine-grained complexity hypotheses about the impossibility of detecting a triangle in edge-linear time, we prove that the k = 9 case does not admit quasi-linear time algorithms. Hence we fully characterize the values of k for which quasi-linear time algorithms exist for polylogarithmic weights on bounded-degree graphs

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions $I_j(x)$ of the first kind, which are Laurent series coefficients of the analytic function family $e^{\frac x2(z+\frac 1z)}$. For every $l\geq1$ we study the family parametrized by $k, n\in\mathbb Z^l$, $k_1>\dots>k_l$, $n_1>\dots>n_l$ of $(l\times l)$-matrix functions formed by the modified Bessel functions of the first kind $a_{ij}(x)=I_{k_j-n_i}(x)$, $i,j=1,\dots,l$. We show that their determinants $f_{k,n}(x)$ are positive for every $l\geq1$, $k,n\in\mathbb Z^l$ as above and $x>0$. The above determinants are closely related to a sequence (indexed by $l$) of families of double confluent Heun equations, which are linear second order differential equations with two irregular singularities, at zero and at infinity. V.M.Buchstaber and S.I.Tertychnyi have constructed their holomorphic solutions on $\mathbb C$ for an explicit class of parameter values and conjectured that they do not exist for other parameter values. They have reduced their conjecture to the second conjecture saying that if an appropriate second similar equation has a polynomial solution, then the first one has no entire solution. They have proved the latter statement under the additional assumption (third conjecture) that $f_{k,n}(x)\neq0$ for $k=(l,\dots,1)$, $n=(l-1,\dots,0)$ and every $x>0$. Our more general result implies all the above conjectures, together with their corollary for the overdamped model of the Josephson junction in superconductivity: the description of adjacency points of phase-lock areas as solutions of explicit analytic equations.Comment: 19 pages, 1 figure. To appear in Nonlinearity. Minor changes. The present version includes additional historical remarks and bibliograph