On quasi-strongly regular graphs

We study the quasi-strongly regular graphs, which are a combinatorial generalization of the strongly regular and the distance regular graphs. Our main focus is on quasi-strongly regular graphs of grade 2. We prove a “spectral gap”-type result for them which generalizes Seidel's well-known formula for the eigenvalues of a strongly regular graph. We also obtain a number of necessary conditions for the feasibility of parameter sets and some structural results. We propose the heuristic principle that the quasi-strongly regular graphs can be viewed as a “lower-order approximation” to the distance regular graphs. This idea is illustrated by extending a known result from the distance-regular case to the quasi-strongly regular case. Along these lines, we propose a number of conjectures and open problems. Finally, we list the all the proper connected quasi-strongly graphs of grade 2 with up to 12 vertices.     

Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets

In this paper, we give a new lifting construction of “hyperbolic” type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to m-ovoids and i-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.     

The local spectra of regular line graphs

The local spectrum of a graph G=(V,E), constituted by the standard eigenvalues of G and their local multiplicities, plays a similar role as the global spectrum when the graph is “seen” from a given vertex. Thus, for each vertex i∈V, the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue λ of G is the sum, extended to all vertices, of its local multiplicities.In this work, using the interpretation of an eigenvector as a charge distribution on the vertices, we compute the local spectrum of the line graph LG in terms of the local spectrum of the regular graph G it derives from. Furthermore, some applications of this result are derived as, for instance, some results about the number of circuits of LG.     

Subgroup regular sets in Cayley graphs

Let Γ be a graph with vertex set V, and let a and b be nonnegative integers. A subset C of V is called an $(a,b)$-regular set in Γ if every vertex in C has exactly a neighbors in C and every vertex in $V?C$ has exactly b neighbors in C. In particular, $(0,1)$-regular sets and $(1,1)$-regular sets in Γ are called perfect codes and total perfect codes in Γ, respectively. A subset C of a group G is said to be an $(a,b)$-regular set of G if there exists a Cayley graph of G which admits C as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group G or any group G of order 4p or pq for some primes p and q, if a nontrivial subgroup H of G is a $(0,1)$-regular set of G, then it must also be an $(a,b)$-regular set of G for any $0?a?|H|?1$ and $0?b?|H|$ such that a is even when $|H|$ is odd. A similar result involving $(1,1)$-regular sets of such groups is also obtained in the paper.     

Block graphs with unique minimum dominating sets

For any graph G a set D of vertices of G is a dominating set, if every vertex vV(G)−D has at least one neighbor in D. The domination number γ(G) is the smallest number of vertices in any dominating set. In this paper, a characterization is given for block graphs having a unique minimum dominating set. With this result, we generalize a theorem of Gunther, Hartnell, Markus and Rall for trees.     

Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counter‐intuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braess's paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erd?s‐Rényi random graphs G (n, p ) with constant edge density , the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of G (n, p ), which might be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 584–611, 2017     

Estimation of spectral gap for Markov chains

The study of the convergent rate (spectral gap) in theL ²-sense is motivated from several different fields: probability, statistics, mathematical physics, computer science and so on and it is now an active research topic. Based on a new approach (the coupling technique) introduced in [7] for the estimate of the convergent rate and as a continuation of [4], [5], [7–9], [23] and [24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper [10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings^[4] is also extended in the paper.Research supported in part by NSFC, Qin Shi Sci & Tech. Foundation and the State Education Commission of China.     

Estimation of spectral gap for elliptic operators

A variational formula for the lower bound of the spectral gap of an elliptic operator is presented in the paper for the first time. The main known results are either recovered or improved. A large number of new examples with sharp estimate are illustrated. Moreover, as an application of the march coupling, the Poincaré inequality with respect to the absolute distribution of the process is also studied.

     

正则图的变换图的谱

设G是一个图,类似全图的定义,可以定义G的8种变换图.如果G是正则图,那么图G的变换图的谱都可以由图G的谱计算得到.     

Bounds on the clique-transversal number of regular graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted Tc(G),is the minimum cardinality of a clique- transversal set in G.In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound.Also,we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.     

Rank-determining sets of metric graphs

A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph Γ is an element of the free abelian group on Γ. The rank of a divisor on a metric graph is a concept appearing in the Riemann-Roch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and Zharkov. We define a rank-determining set of a metric graph Γ to be a subset A of Γ such that the rank of a divisor D on Γ is always equal to the rank of D restricted on A. We show constructively in this paper that there exist finite rank-determining sets. In addition, we investigate the properties of rank-determining sets in general and formulate a criterion for rank-determining sets. Our analysis is based on an algorithm to derive the v₀-reduced divisor from any effective divisor in the same linear system.     

Tight bounds for eternal dominating sets in graphs

The eternal domination number of a graph is the number of guards needed at vertices of the graph to defend the graph against any sequence of attacks at vertices. We consider the model in which at most one guard can move per attack and a guard can move across at most one edge to defend an attack. We prove that there are graphs G for which , where γ_∞(G) is the eternal domination number of G and α(G) is the independence number of G. This matches the upper bound proved by Klostermeyer and MacGillivray.     

Infinitely many pairs of cospectral integral regular graphs

A graph G is called integral if all the eigenvalues of the adjacency matrix A(G) of G are integers. In this paper, the graphs G ₄(a, b) and G ₅(a, b) with 2a+6b vertices are defined. We give their characteristic polynomials from matrix theory and prove that the (n+2)-regular graphs G ₄(n, n+2) and G ₅(n, n+2) are a pair of non-isomorphic connected cospectral integral regular graphs for any positive integer n.     

Some results on the spectral radii of bicyclic graphs

A bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. Let Δ(G) and ρ(G) denote the maximum degree and the spectral radius of a graph G, respectively. Let B(n) be the set of bicyclic graphs on n vertices, and B(n,Δ)={G∈B(n)∣Δ(G)=Δ}. When Δ≥(n+3)/2 we characterize the graph which alone maximizes the spectral radius among all the graphs in B(n,Δ). It is also proved that for two graphs G₁ and G₂ in B(n), if Δ(G₁)>Δ(G₂) and Δ(G₁)≥⌈7n/9⌉+9, then ρ(G₁)>ρ(G₂).     

Invariant Gaussian processes and independent sets on regular graphs of large girth

We prove that every 3‐regular, n‐vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d‐regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue . We show that such processes can be approximated by i.i.d. factors provided that . We then use these approximations for to produce factor of i.i.d. independent sets on regular trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 284–303, 2015