The spectral characterization of graphs of index less than 2 with no path as a component

A graph is said to be determined by the adjacency and Laplacian spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency and Laplacian spectrum, respectively. It is known that connected graphs of index less than 2 are determined by their adjacency spectrum. In this paper, we focus on the problem of characterization of DS graphs of index less than 2. First, we give various infinite families of cospectral graphs with respect to the adjacency matrix. Subsequently, the results will be used to characterize all DS graphs (with respect to the adjacency matrix) of index less than 2 with no path as a component. Moreover, we show that most of these graphs are DS with respect to the Laplacian matrix.     

Graphs for small multiprocessor interconnection networks

Let D be the diameter of a graph G and let λ₁ be the largest eigenvalue of its (0, 1)-adjacency matrix. We give a proof of the fact that there are exactly 69 non-trivial connected graphs with (D + 1)λ₁ ? 9. These 69 graphs all have up to 10 vertices and were recently found to be suitable models for small multiprocessor interconnection networks. We also examine the suitability of integral graphs to model multiprocessor interconnection networks, especially with respect to the load balancing problem. In addition, we classify integral graphs with small values of (D + 1)λ₁ in connection with the load balancing problem for multiprocessor systems.     

Distance spectra and distance energy of integral circulant graphs

The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix D. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs (G₁,G₂) of integral circulant graphs with equal distance energy - in the first family G₁ is subgraph of G₂, while in the second family the diameter of both graphs is three.     

On the asymptotic behavior of graphs determined by their generalized spectra

In Wang and Xu (2006) [15] and [16] the authors introduced a family of graphs H_n and gave some methods for finding graphs among this family that are determined by their generalized spectra. This paper is a continuation of our previous work. We further show that almost all graphs in H_n are determined by their generalized spectra. This gives some evidences for the conjecture that almost all graphs are determined by their generalized spectra.     

Commutativity of the adjacency matrices of graphs

We say that two graphs G₁ and G₂ with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph K_n,n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n−1 linearly independent commuting adjacency matrices of size n; and if this bound occurs, then there exists a Hadamard matrix of order n. Finally, we determine the centralizers of some families of graphs.     

Spectral characterization of line graphs of starlike trees

Two graphs are said to be A-cospectral if they have the same adjacency spectrum. A graph G is said to be determined by its adjacency spectrum if there is no other non-isomorphic graph A-cospectral with G. A tree is called starlike if it has exactly one vertex of degree greater than 2. In this article, we prove that the line graphs of starlike trees with maximum degree at least 12 are determined by their adjacency spectra.     

Almost all steinhaus graphs have diameter 2

A Steinhaus graph is a graph with n vertices whose adjacency matrix (a_{i, j}) satisfies the condition that a_{i, j} ? a_{a--1, j--1} + a _{i--1, j} (mod 2) for each 1 < i < j ≤ n. It is clear that a Steinhaus graph is determined by its first row. In [3] Bringham and Dutton conjecture that almost all Steinhaus graphs have diameter 2. That is, as n approaches infinity, the ratio of the number of Steinhaus graphs with n vertices having diameter 2 to the total number of Steinhaus graphs approaches 1. Here we prove Bringham and Dutton's conjecture.     

On spectral characterizations and embeddings of graphs

The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G. In this paper all graphs with λ(G) greater than ?2 are characterized. Such a graph is a generalized line graph of the form L(T;1,0,…,0), L(T), L(H), where T is a tree and H is unicyclic with an odd cycle, or is one of 573 graphs that arise from the root system E₈. If G is regular with λ(G)>?2, then Gis a clique or an odd circuit. These characterizations are used for embedding problems; λ_R(H) = sup{λ(G)z.sfnc;HinG; Gregular}. H is an odd circuit, a path, or a complete graph iff λ_R(H)> ?2. For any other line graph H, λ_R(H) = ?2. A similar result holds for complete multipartite graphs.     

On strongly regular graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>1</Subscript> &lt; 24

Let γ be a connected edge-regular graph with parameters (v, k, λ), and let b ₁ = k?λ?1. It is well known that, if b ₁ = 1, then Γ is either a polygon or a complete multipartite graph with parts of order 2. Graphs with b ₁ ≤ 4 were classified earlier. The investigation of graphs even in the case b ₁ = 5 involves great difficulties. However, for strongly regular graphs, the situation is much simpler. In this paper, we classify strongly regular graphs with b ₁ < 24.     

Graph polynomials from principal pivoting

The recursive computation of the interlace polynomial introduced by Arratia, Bollobás and Sorkin is defined in terms of a new pivoting operation on undirected simple graphs. In this paper, we interpret the new pivoting operation on graphs in terms of standard pivoting (on matrices). Specifically, we show that, up to swapping vertex labels, Arratia et al.'s pivoting operation on a graph is equivalent to a principal pivot transform on the graph's adjacency matrix, provided that all computations are performed in the Galois field F₂. Principal pivoting on adjacency matrices over F₂ has a natural counterpart on isotropic systems. Thus, our view of the interlace polynomial is closely related to the one by Aigner and van der Holst.The observations that adjacency matrices of undirected simple graphs are skew-symmetric in F₂ and that principal pivoting preserves skew-symmetry in all fields suggest to extend Arratia et al.'s pivoting operation to fields other than F₂. Thus, the interlace polynomial extends to polynomials on gain graphs, namely bidirected edge-weighted graphs whereby reversed edges carry non-zero weights that differ only by their sign. Extending a proof by Aigner and van der Holst, we show that the extended interlace polynomial can be represented in a non-recursive form analogous to the non-recursive form of the original interlace polynomial, i.e., the Martin polynomial.For infinite fields it is shown that the extended interlace polynomial does not depend on the (non-zero) gains, as long as they obey a non-singularity condition. These gain graphs are all supported by a single undirected simple graph. Thus, a new graph polynomial is defined for undirected simple graphs. The recursive computation of the new polynomial can be done such that all ends of the recursion correspond to independent sets. Moreover, its degree equals the independence number. However, the new graph polynomial is different from the independence polynomial.     

On a class of critical singular quasilinear elliptic problem with indefinite weights

In this paper, the eigenvalue problem for a class of quasilinear elliptic equations involving critical potential and indefinite weights is investigated. We obtain the simplicity, strict monotonicity and isolation of the first eigenvalue λ₁. Furthermore, because of the isolation of λ₁, we prove the existence of the second eigenvalue λ₂. Then, using the Trudinger-Moser inequality, we obtain the existence of a nontrivial weak solution for a class of quasilinear elliptic equations involving critical singularity and indefinite weights in the case of 0<λ<λ₁ by the Mountain Pass Lemma, and in the case of λ₁≤λ<λ₂ by the Linking Argument Theorem.     

On the spread of the spectrum of a graph

Some new upper bounds and lower bounds are obtained for the spread λ₁−λ_n of the eigenvalues λ₁≥λ₂≥?≥λ_n of the adjacency matrix of a simple graph.     

Spectral characterizations of sandglass graphs

The sandglass graph is obtained by appending a triangle to each pendant vertex of a path. It is proved that sandglass graphs are determined by their adjacency spectra as well as their Laplacian spectra.     

Existence of Dλ-cycles and Dλ-paths

A cycle of C of a graph G is called a D_λ-cycle if every component of G ? V(C) has order less than λ. A D_λ-path is defined analogously. In particular, a D₁-cycle is a hamiltonian cycle and a D₁-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a D_λ-cycle or D_λ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way.     

An accessibility theorem for infinite graph minors

We prove the following recent conjecture of Halin. Let Γ₀ be the class of all graphs, and for every ordinal μ > 0 let Γ_μ be the class of all graphs containing infinitely many disjoint connected graphs from Γ_λ, for every λ < μ. Then a graph lies in all these classes Γ_μ if and only if it contains a subdivision of the infinite binary tree. Published by John Wiley & Sons, Inc., 2000 J Graph Theory 35: 273–277, 2000     

On a signless Laplacian spectral characterization of -shape trees

Let M be an associated matrix of a graph G (the adjacency, Laplacian and signless Laplacian matrix). Two graphs are said to be cospectral with respect to M if they have the same M spectrum. A graph is said to be determined by M spectrum if there is no other non-isomorphic graph with the same spectrum with respect to M. It is shown that T-shape trees are determined by their Laplacian spectra. Moreover among them those are determined by their adjacency spectra are characterized. In this paper, we identify graphs which are cospectral to a given T-shape tree with respect to the signless Laplacian matrix. Subsequently, T-shape trees which are determined by their signless Laplacian spectra are identified.     

Maximal reflexive cacti with four cycles: The approach via Smith graphs

Cacti, or treelike graphs, are graphs whose all cycles are mutually edge-disjoint. Graphs with the property are called reflexive graphs, where λ₂ is the second largest eigenvalue of the corresponding (0, 1)-adjacency matrix. The property is a hereditary one, i.e. all induced subgraphs of a reflexive graph are also reflexive. This is why we represent reflexive graphs through the maximal graphs within a given class (such as connected cacti with a fixed number of cycles). In previous work we have determined all maximal reflexive cacti with four cycles whose cycles do not form a bundle and pointed out the role of so-called pouring of Smith graphs in their construction. In this paper, besides pouring, we show several other patterns of the appearance of Smith trees in those constructions. These include splitting of a Smith tree, adding an edge to a Smith tree and then splitting of the resulting graph, identifying two vertices of a Smith tree and then splitting the resulting graph. Our results show that the presence of Smith trees is evident in all such maximal reflexive cacti with four cycles and that in most of them Smith graphs appear in the described way.     

On the spectral radius of ‡-shape trees

Let A(G) be the adjacency matrix of G. The characteristic polynomial of the adjacency matrix A is called the characteristic polynomial of the graph G and is denoted by φ(G, λ) or simply φ(G). The spectrum of G consists of the roots (together with their multiplicities) λ ₁(G) ? λ ₂(G) ? … ? λ _n (G) of the equation φ(G, λ) = 0. The largest root λ ₁(G) is referred to as the spectral radius of G. A ?-shape is a tree with exactly two of its vertices having maximal degree 4. We will denote by G(l ₁, l ₂, … l ₇) (l ₁ ? 0, l _i ? 1, i = 2, 3, …, 7) a ?-shape tree such that $G\left( {l_1 ,l_2 , \ldots l_7 } \right) - u - v = P_{l_1 } \cup P_{l_2 } \cup \ldots P_{l_7 }$ , where u and v are the vertices of degree 4. In this paper we prove that ${{3\sqrt 2 } \mathord{\left/ {\vphantom {{3\sqrt 2 } 2}} \right. \kern-0em} 2} < \lambda _1 \left( {G\left( {l_1 ,l_2 , \ldots l_7 } \right)} \right) < {5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}$ .     

On the edge-connectivity and restricted edge-connectivity of a product of graphs

The product graph G_m*G_p of two given graphs G_m and G_p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters (λ and λ^′, edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G_m, G_p of minimum degrees δ(G_m), δ(G_p), respectively, we show that λ(G_m*G_p) is lower bounded by both δ(G_m)+λ(G_p) and δ(G_p)+λ(G_m), an improvement of what is known for the edge-connectivity of G_m×G_p.     

Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices

Let Ω⊂R² be a simply connected domain, let ω be a simply connected subdomain of Ω, and set A=Ω?ω. Suppose that J is the class of complex-valued maps on the annular domain A with degree 1 both on ∂Ω and on ∂ω. We consider the variational problem for the Ginzburg-Landau energy Eλ among all maps in J. Because only the degree of the map is prescribed on the boundary, the set J is not necessarily closed under a weak H¹-convergence. We show that the attainability of the minimum of Eλ over J is determined by the value of cap(A)—the H¹-capacity of the domain A. In contrast, it is known, that the existence of minimizers of Eλ among the maps with a prescribed Dirichlet boundary data does not depend on this geometric characteristic. When cap(A)?π (A is either subcritical or critical), we show that the global minimizers of Eλ exist for each λ>0 and they are vortexless when λ is large. Assuming that λ→∞, we demonstrate that the minimizers of Eλ converge in H¹(A) to an S¹-valued harmonic map which we explicitly identify. When cap(A)<π (A is supercritical), we prove that either (i) there is a critical value λ₀ such that the global minimizers exist when λ<λ₀ and they do not exist when λ>λ₀, or (ii) the global minimizers exist for each λ>0. We conjecture that the second case never occurs. Further, for large λ, we establish that the minimizing sequences/minimizers in supercritical domains develop exactly two vortices—a vortex of degree 1 near ∂Ω and a vortex of degree −1 near ∂ω.