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In his survey Beyond graph energy: Norms of graphs and matrices (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph, respectively. We show that these graphs have at most two nonzero distinct absolute eigenvalues and investigate the proposed problems organizing our study according to the type of spectrum they can have. In most cases all graphs are characterized. Infinite families of graphs are given otherwise. We also show that all graphs satifying the properties required in the problems are integral, except for complete bipartite graphs $K_{p,q}$ and disconnected graphs with a connected component $K_{p,q}$, where $pq$ is not a perfect square.
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We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching.
A signed graph is a pair $(G,Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V={1,ldots,n}$ and $Sigmasubseteq E$. The edges in $Sigma$ are called odd and the other edges of $E$ even. By $S(G,Sigma)$ we denote the set of all symmetric $ntimes n$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are adjacent and connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are adjacent and connected by only odd edges, $a_{i,j}in mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i ot=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} in mathbb{R}$ for all vertices $i$. The parameters $M(G,Sigma)$ and $xi(G,Sigma)$ of a signed graph $(G,Sigma)$ are the largest nullity of any matrix $Ain S(G,Sigma)$ and the largest nullity of any matrix $Ain S(G,Sigma)$ that has the Strong Arnold Hypothesis, respectively. In a previous paper, we gave a characterization of signed graphs $(G,Sigma)$ with $M(G,Sigma)leq 1$ and of signed graphs with $xi(G,Sigma)leq 1$. In this paper, we characterize the $2$-connected signed graphs $(G,Sigma)$ with $M(G,Sigma)leq 2$ and the $2$-connected signed graphs $(G,Sigma)$ with $xi(G,Sigma)leq 2$.
In this paper, we introduce the concepts of the plain eigenvalue, the main-plain index and the refined spectrum of graphs. We focus on the graphs with two main and two plain eigenvalues and give some characterizations of them.
Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges, and so we can ask, for example, how likely is it for the process to produce a graph with $k$ trees. We look at all graphs which can produce at most two trees in this process and determine the probabilities of having either one or two trees. From this we construct infinite families of graphs which are non-isomorphic but produce the same probabilities.
We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are largest. For $k e 2,3,7,57$, we show an upper bound $nleq k^2-k+1$, with equality if and only if there exists a finite projective plane of order $k-1$ that admits a polarity.
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