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Abstract

Let A(G) and D(G) be the adjacency matrix and the degree matrix of a graph G, respectively. For any real α ∈ [0, 1], we consider Aα (G) = αD(G) + (1 − α)A(G) as a graph matrix, whose largest eigenvalue is called the Aα -spectral radius of G. We first show that the smallest limit point for the Aα -spectral radius of graphs is 2, and then we characterize the connected graphs whose Aα -spectral radius is at most 2. Finally, we show that all such graphs, with four exceptions, are determined by their Aα -spectra.

Abstract

The eigenvalues of a graph are those of its adjacency matrix. Recently, Cioabă, Haemers and Vermette characterized all graphs with all but two eigenvalues equal to 2 and 0. In this article, we extend their result by characterizing explicitly all graphs with all but two eigenvalues in the interval [2, 0]. Also, we determine among them those that are determined by their spectrum.

Abstract

Given a graph G, its adjacency matrix A(G) and its diagonal matrix of vertex degrees D(G), consider the matrix Aα (G) = αD(G) + (1 − α)A(G), where α ∈ [0, 1). The Aα -spectrum of G is the multiset of eigenvalues of Aα (G) and these eigenvalues are the α-eigenvalues of G. A cluster in G is a pair of vertex subsets (C, S), where C is a set of cardinality |C| ≥ 2 of pairwise co-neighbor vertices sharing the same set S of |S| neighbors. Assuming that G is connected and it has a cluster (C, S), G(H) is obtained from G and an r-regular graph H of order |C| by identifying its vertices with the vertices in C, eigenvalues of Aα (G) and Aα (G(H)) are deduced and if Aα (H) is positive semidefinite, then the i-th eigenvalue of Aα (G(H)) is greater than or equal to i-th eigenvalue of Aα (G). These results are extended to graphs with several pairwise disjoint clusters (C 1, S 1), . . . , (Ck, Sk). As an application, the effect on the energy, α-Estrada index and α-index of a graph G with clusters when the edges of regular graphs are added to G are analyzed. Finally, the Aα-spectrum of the corona product G ◦ H of a connected graph G and a regular graph H is determined.

Abstract

Let A be a real symmetric matrix. If after we delete a row and a column of the same index, the nullity increases by one, we call that index a P-vertex of A. When A is an n × n singular acyclic matrix, it is known that the maximum number of P-vertices is n − 2. If T is the underlying tree of A, we will show that for any integer number k{0, 1, . . . , n − 2}, there is a (singular) matrix whose graph is T and with k P-vertices. We will provide illustrative examples.

Abstract

In this paper our focus is on regular signed graphs with exactly 3 (distinct) eigenvalues. We establish certain basic results; for example, we show that they are walk-regular. We also give some constructions and determine all the signed graphs with 3 eigenvalues, under the constraint that they are either signed line graphs or have vertex degree 3. We also report our result of computer search on those with at most 10 vertices.

Abstract

Twin vertices of a graph have the same open neighbourhood. If they are not adjacent, then they are called duplicates and contribute the eigenvalue zero to the adjacency matrix. Otherwise they are termed co-duplicates, when they contribute 1 as an eigenvalue of the adjacency matrix. On removing a twin vertex from a graph, the spectrum of the adjacency matrix does not only lose the eigenvalue 0 or 1. The perturbation sends a rippling effect to the spectrum. The simple eigenvalues are displaced. We obtain a closed formula for the characteristic polynomial of a graph with twin vertices in terms of two polynomials associated with the perturbed graph. These are used to obtain estimates of the displacements in the spectrum caused by the perturbation.

Abstract

We consider the problem of maximizing the distance spectral radius and a slight generalization thereof among all trees with some prescribed degree sequence. We prove in particular that the maximum of the distance spectral radius has to be attained by a caterpillar for any given degree sequence. The same holds true for the terminal distance matrix. Moreover, we consider a generalized version of the reverse distance matrix and also study its spectral radius for trees with given degree sequence. We prove that the spectral radius is always maximized by a greedy tree. This implies several corollaries, among them a “reversed” version of a conjecture of Stevanović and Ilić. Our results parallel similar theorems for the Wiener index and other invariants.

Abstract

For 0 ≤ α ---lt--- 1 and a uniform hypergraph G, the α-spectral radius of G is the largest H-eigenvalue of αD(G)+(1−α)A(G), where D(G) and A(G) are the diagonal tensor of degrees and the adjacency tensor of G, respectively. We give upper bounds for the α-spectral radius of a uniform hypergraph, propose some transformations that increase the α-spectral radius, and determine the unique hypergraphs with maximum α-spectral radius in some classes of uniform hypergraphs.

Abstract

Let G be a connected r-regular graph (r ---gt--- 3) of order n with a tree of order t as a star complement for an eigenvalue µ ∉ {−1, 0}. It is shown that n ≤ 1/2 (r + 1)t − 2. Equality holds when G is the complement of the Clebsch graph (with µ = 1, r = 5, t = 6, n = 16).

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