## 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 *A _{α}* (

*G*) = α

*D*(

*G*) + (1

*− α*)

*A*(

*G*) as a graph matrix, whose largest eigenvalue is called the

*A*-spectral radius of

_{α}*G*. We first show that the smallest limit point for the

*A*-spectral radius of graphs is 2, and then we characterize the connected graphs whose

_{α}*A*-spectral radius is at most 2. Finally, we show that all such graphs, with four exceptions, are determined by their

_{α}*A*-spectra.

_{α}## 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 *A _{α}* (

*G*) = α

*D*(

*G*) + (1

*− α*)

*A*(

*G*), where α ∈ [0, 1). The

*A*-spectrum of

_{α}*G*is the multiset of eigenvalues of

*A*(

_{α}*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

*A*(

_{α}*G*) and

*A*(

_{α}*G*(

*H*)) are deduced and if

*A*(

_{α}*H*) is positive semidefinite, then the

*i*-th eigenvalue of

*A*(

_{α}*G*(

*H*)) is greater than or equal to

*i*-th eigenvalue of

*A*(

_{α}*G*). These results are extended to graphs with several pairwise disjoint clusters (

*C*

_{1},

*S*

_{1}), . . . , (

*C*,

_{k}*S*). As an application, the effect on the energy, α-Estrada index and α-index of a graph

_{k}*G*with clusters when the edges of regular graphs are added to

*G*are analyzed. Finally, the

*A*-spectrum of the corona product

_{α}*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).