Observations on the Lov\'asz θ\theta-Function, Graph Capacity, Eigenvalues, and Strong Products

This paper provides new observations on the Lov\'{a}sz $\theta$-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lov\'{a}sz, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lov\'{a}sz $\theta$-function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lov\'{a}sz $\theta$-function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a $k$-fold strong power of a regular graph is compared to the Alon--Boppana bound; under a certain condition, the new bound is superior in its exponential growth rate (in $k$). Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and the Lov\'{a}sz $\theta$-function of each factor. The utility of these bounds is exemplified, leading in some cases to an exact determination of the chromatic numbers of strong products or strong powers of graphs. The present research paper is aimed to have tutorial value as well.Comment: electronic links to references were added in version 2; Available at https://www.mdpi.com/1099-4300/25/1/10

Some new considerations about double nested graphs

In the set of all connected graphs with fixed order and size, the graphs with maximal index are nested split graphs, also called threshold graphs. It was recently (and independently) observed in [F.K.Bell, D. Cvetkovi´c, P. Rowlinson, S.K. Simi´c, Graphs for which the largest eigenvalue is minimal, II, Linear Algebra Appl. 429 (2008)] and [A. Bhattacharya, S. Friedland, U.N. Peled, On the first eigenvalue of bipartite graphs, Electron. J. Combin. 15 (2008), #144] that double nested graphs, also called bipartite chain graphs, play the same role within class of bipartite graphs. In this paper we study some structural and spectral features of double nested graphs. In studying the spectrum of double nested graphs we rather consider some weighted nonnegative matrices (of significantly less order) which preserve all positive eigenvalues of former ones. Moreover, their inverse matrices appear to be tridiagonal. Using this fact we provide several new bounds on the index (largest eigenvalue) of double nested graphs, and also deduce some bounds on eigenvector components for the index. We conclude the paper by examining the questions related to main versus non-main eigenvalues

Laplacian spread of graphs: lower bounds and relations with invariant parameters

The spread of an $n\times n$ complex matrix $B$ with eigenvalues $\beta _{1},\beta _{2},\ldots ,\beta _{n}$ is defined by \begin{equation*} s\left( B\right) =\max_{i,j}\left\vert \beta _{i}-\beta _{j}\right\vert , \end{equation*}% where the maximum is taken over all pairs of eigenvalues of $B$. Let $G$ be a graph on $n$ vertices. The concept of Laplacian spread of $G$ is defined by the difference between the largest and the second smallest Laplacian eigenvalue of $G$. In this work, by combining old techniques of interlacing eigenvalues and rank $1$ perturbation matrices new lower bounds on the Laplacian spread of graphs are deduced, some of them involving invariant parameters of graphs, as it is the case of the bandwidth, independence number and vertex connectivity

Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs

The Laplacian spectra are the eigenvalues of Laplacian matrix L(G)=D(G)-A(G), where D(G) and A(G) are the diagonal matrix of vertex degrees and the adjacency matrix of a graph G, respectively, and the spectral radius of a graph G is the largest eigenvalue of A(G). The spectra of the graph and corresponding eigenvalues are closely linked to the molecular stability and related chemical properties. In quantum chemistry, spectral radius of a graph is the maximum energy level of molecules. Therefore, good upper bounds for the spectral radius are conducive to evaluate the energy of molecules. In this paper, we first give several sharp upper bounds on the adjacency spectral radius in terms of some invariants of graphs, such as the vertex degree, the average 2-degree, and the number of the triangles. Then, we give some numerical examples which indicate that the results are better than the mentioned upper bounds in some sense. Finally, an upper bound of the Nordhaus-Gaddum type is obtained for the sum of Laplacian spectral radius of a connected graph and its complement. Moreover, some examples are applied to illustrate that our result is valuable