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Let be a finite-dimensional Hilbert space and be a finite frame for . For , we associate a simple graph denoted by and called frame graph, with all elements of as vertices, and two distinct vertices are adjacent when the inner product of them is nonzero. The associated graph is called a tight frame graph if the mentioned frame is tight. In this paper, we study some basic properties of and investigate the relation between the frame-theoretic properties of and the graph-theoretic properties of . For example, it is shown that trees on more than two vertices and cycles on more than five vertices are not tight frame graphs while complete graphs, complete graphs minus a single edge and the join of each graph with itself are tight frame graphs. [ABSTRACT FROM AUTHOR]

In this paper we focus on connected signed graphs of fixed number of vertices, positive edges and negative edges that maximize the largest eigenvalue (also called the index) of their adjacency matrix. In the first step we determine these signed graphs in the set of signed generalized theta graphs. Concerning the general case, we use the eigenvector techniques for getting some structural properties of resulting signed graphs. In particular, we prove that positive edges induce nested split subgraphs, while negative edges induce double nested signed subgraphs.We observe that our concept can be applied when considering balancedness of signed graphs (the property that is extensively studied in both mathematical and non-mathematical context). [ABSTRACT FROM AUTHOR]

A weighted Bethe graphBis obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a familyFof graphs. The operation of identifying the root vertex of each ofrweighted Bethe graphs to the vertices of a connected graphof orderris introduced as the-concatenation of a family ofrweighted Bethe graphs. It is shown that the Laplacian eigenvalues (whenFhas arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs inFare all regular) of the-concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered. [ABSTRACT FROM PUBLISHER]

Graphs with least eigenvalue greater than or equal toare to a big extent studied by Hoffman and other authors from the early beginning of the spectral graph theory. Most of these results are summarized in the monograph [Cvetković D, Rowlinson P, Simić S. Spectral generalizations of line graphs, on graphs with least eigenvalue, Cambridge University Press, 2004], and the survey paper [Cvetković D, Rowlinson P, Simić S. Graphs with least eigenvalue: ten years on, Linear Algebra Appl. 2015;484:504–539] which is aimed to cover the next 10 years since their monograph appeared. Here, we add some further results. Among others, we identify graphs whose least eigenvalue is greater than, but closest towithin the graphs of fixed order. Some consequences of these considerations are found in the context of the highest occupied molecular orbital–lowest unoccupied molecular orbital invariants. [ABSTRACT FROM AUTHOR]

A graphis nonsingular if its adjacency matrixis nonsingular. A nonsingular graphis said to have an inverseifis signature similar to a nonnegative matrix. An-graphis a bipartite graph with a unique perfect matching. We introduce the ‘even’ness of a nonmatching edge and show that under certain conditions a connected-graph has an inverse if and only ifis bipartite, whereis the set of all even nonmatching edges. This extends some known results, providing us with a larger class of graphs possessing inverses. [ABSTRACT FROM PUBLISHER]

A signed graphconsists of an underlying graphand a sign function, which assigns each edge ofa sign, either positive or negative. The adjacency matrix of a signed graphis defined aswith, whereis the adjacency matrix of the underlying graph. The rank ofis defined to be the rank of. In this paper, we characterize the signed graphs with rankor, and characterize the signed planar graphs with rank 4. [ABSTRACT FROM PUBLISHER]

Letandbe two vertex disjoint graphs of ordersandrespectively, whereandLetbe a specified edge ofsuch thatis isomorphic toLetbe a subset of the edge set ofand letdenote the subgraph ofinduced byLetbe the graph obtained by taking one copy ofandvertex disjoint copies ofand then pasting the edgein theth copy ofwith the edgewhereThen the copies of the graphthat are pasted to the edgesare called as edge-pockets, and we sayis a graph withedge-pockets. In this article, we prove some results describing the Laplacian (resp. adjacency) spectrum ofusing the Laplacian (resp. adjacency) spectra ofandThe complete Laplacian (resp. adjacency) spectrum ofis also described in some particular cases. As an application, we show that these results enable us to construct infinitely many pairs of Laplacian (resp. adjacency) cospectral graphs. [ABSTRACT FROM AUTHOR]

A block graph is a graph in which every block is a complete graph. Letbe a block graph and letbe the adjacency matrix of. We first obtain a formula for the determinant ofover reals. It is shown thatis nonsingular overif and only if the removal of any vertex fromproduces a graph with exactly one odd component. A formula for the inverse ofoveris obtained, whenever it exists. We obtain some results for the adjacency matrices over, of claw-free block graphs, which are the same as the line graphs of trees, and for the adjacency matrices of flowers, which are block graphs with only one cut-vertex. [ABSTRACT FROM AUTHOR]

Two graphs are said to be A-cospectral if they have the same adjacency spectrum. A graphGis said to be determined by its adjacency spectrum if there is no other non-isomorphic graph A-cospectral withG. 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. [ABSTRACT FROM AUTHOR]

The nullity η(G) of a graph G is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. If η(G)= 1, then the core of G is the subgraph induced by the vertices associated with the nonzero entries of the kernel eigenvector. The set of vertices which are not in the core is the periphery of G. A graph G with nullity one is minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. An ∞-graph ∞(p, l, q) is a graph obtained by joining two vertex-disjoint cycles Cp and Cq by a path of length l≥ 0. Let B* be the class of bicyclic graphs with an ∞-graph as an induced subgraph. In this article, we characterize the graphs in B* which are minimal configurations. [ABSTRACT FROM AUTHOR]

The signless Laplacian matrix of a graph is defined to be the sum of its adjacency matrix and degree matrix. Let [image omitted] be the set of all the connected graphs of order n and diameter d and Gn,k the set of all connected graphs with order n and k cut vertices. In this article, we determine the graphs that have the maximal signless Laplacian spectral radius and give the upper bounds of graphs in these two sets. [ABSTRACT FROM AUTHOR]

In this note we show how to construct two distinct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices by 'unfolding' a base bipartite graph in two different ways. [ABSTRACT FROM AUTHOR]

A graph is singular of nullity η if zero is an eigenvalue of its adjacency matrix with multiplicity η. If η( G )=1, then the core of G is the subgraph induced by the vertices associated with the non-zero entries of the zero-eigenvector. A connected subgraph of G with the least number of vertices and edges, that has nullity one and the same core as G , is called a minimal configuration. A subdivision of a graph G is obtained by inserting a vertex on every edge of G . We review various properties of minimal configurations. In particular, we show that a minimal configuration is a tree if and only if it is a subdivision of some other tree. [ABSTRACT FROM AUTHOR]