Zoran Stanić

Q-integral graphs with edge-degrees at most five

We consider the problem of determining the Q-integral graphs, i.e. the graphs with integral signless Laplacian spectrum. We find all such graphs with maximum edge-degree 4, and obtain only partial results for the next natural case, with maximum edge-degree 5.

On some forests determined by their Laplacian or signless Laplacian spectrum

We consider the class of graphs whose each component is either a proper subgraph of some Smith graphs, or belongs to a precized subset of Smith graphs. We classify the graphs from the considered class into those which are determined, or not determined, by Laplacian, or signless Laplacian spectrum.

On regular graphs and coronas whose second largest eigenvalue does not exceed 1

We characterize all regular graphs whose second largest eigenvalue does not exceed 1. In the sequel, we determine all coronas, different from cones, with the same property. Some results and examples regarding unsolved cases are also given.

CONTROLLABLE GRAPHS WITH LEAST EIGENVALUE AT LEAST 2

Connected graphs whose eigenvalues are distinct and main are called controllable graphs in view of certain applications in control theory. We give some general characterizations of the controllable graphs whose least eigenvalue is bounded from below by 2; in particular, we determine all the controllable exceptional graphs. We also investigate the controllable graphs whose second largest eigenvalue does not exceed 1.

Spectral determination of graphs whose components are paths and cycles

We consider the class of graphs each of whose components is either a path or a cycle. We classify the graphs from the class considered into those which are determined and those which are not determined by the adjacency spectrum. In addition, we compare the result with the corresponding results for the Laplacian and the signless Laplacian spectra. It turns out that the signless Laplacian spectrum performs the best, confirming some expectations from the literature.

Graphs with small spectral gap

It is conjectured that connected graphs with given number of vertices and minimum spectral gap (i.e., the difference between their two largest eigenvalues) are double kite graphs. The conjecture is confirmed for connected graphs with at most 10 vertices, and, using variable neighbourhood metaheuristic, there is evidence that it is true for graphs with at most 15 vertices. Several spectral properties of double kite graphs are obtained, including the equations for their first two eigenvalues. No counterexamples to the conjecture are obtained. Some numerical computations and comparisons that indicate its correctness are also given. Next, 3 lower and 3 upper bounds on spectral gap are derived, and some spectral and structural properties of the graphs that minimize the spectral gap are given. At the end, it is shown that in connected graphs any double kite graph has a unique spectrum.

The polynomial reconstruction of unicyclic graphs is unique

We consider the problem of reconstructing the characteristic polynomial of a graph G from its polynomial deck, i.e. the collection of characteristic polynomials of its vertex-deleted subgraphs. Here we provide a positive solution for all unicyclic graphs.

Further results on controllable graphs

Connected graphs whose eigenvalues are mutually distinct and main are called controllable graphs. In recent work their relevance in control theory is recognized, and a number of theoretical and computational results are obtained. In this paper, some criteria for non-controllability of graphs are considered, and certain constructions of controllable graphs are given. Controllable graphs whose index does not exceed a given constant (close to 2.0366) are limited as part of two specific families of trees, and controllable graphs with extremal diameter are discussed. Some computational results are presented, along with corresponding theoretical observations.

Sharp spectral inequalities for connected bipartite graphs with maximal Q-index

The Q -index of a simple graph is the largest eigenvalue of its signless Laplacian. As for the adjacency spectrum, we will show that in the set of connected bipartite graphs with fixed order and size, the bipartite graphs with maximal Q -index are the double nested graphs. We provide a sequence of (in)equalities regarding the principal eigenvector of the signless Laplacian of double nested graphs and apply these results to obtain some lower and upper bounds for their Q -index. In the end, we give some computational results in order to compare these bounds.

Some Star Complements for the Second Largest Eigenvalue of a Graph

The star complement technique is a spectral tool recently developed for constructing some bigger graphs from their smaller parts, called star complements. The most frequently, the implementation of those technique requires using the computers. Therefrom, we develop an SCL (star complement library) – the set of programs providing easy and quick implementation of those technique. Here, we present the facilities of SCL. In further, we determine some star complements for 1 or (√5 – 1)/2 as the second largest eigenvalue of a graph. Finally, using the SCL, we consider the maximal extensions of the star complements obtained.