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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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