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Let (G, w) be a weighted graph with the adjacency matrix A. The group inverse of (G, w), denoted by (G # , w #) is the weighted graph with the weight w # (v i v j) of an edge v i v j in G # is defined as the ijth entry of A # , the group inverse of A. We study the group inverse of singular weighted trees. It is shown that if (T, w) is a singular weighted tree, then (T # , w #) is again a weighted tree if and only if (T, w) is a star tree, which in turn holds if and only if (T # , w #) is graph isomorphic to (T, w). A new class T w of weighted trees is introduced and studied here. It is shown that the group inverse of the adjacency matrix of a positively weighted tree in T w is signature similar to a non-negative matrix. [ABSTRACT FROM AUTHOR]

We consider particular weighted directed graphs with edges having colour red, blue or green such that each red edge has weight 1, each blue edge has weight $ -1 $ − 1 and each green edge has weight $ \mathrm {i} $ i (the imaginary unit). Such a directed graph is called a 3-coloured digraph (for short, a 3-CD). Every mixed graph and every signed graph can be interpreted as a 3-CD. We first study some structural properties of 3-CDs by means of the eigenvalues of the adjacency matrix. In particular, we give spectral criteria for singularity of such digraphs. Second, we consider weight-symmetric 3-CDs, i.e. those 3-CDs that are switching isomorphic to their negation. It follows that the class of weight-symmetric 3-CDs is included in the class of 3-CDs whose spectrum is symmetric (with respect to the origin). We give some basic properties and several constructions of weight-symmetric 3-CDs and establish constructions of 3-CDs which have symmetric spectrum but are not weight-symmetric. [ABSTRACT FROM AUTHOR]

Twin vertices in simple unweighted graphs are vertices that have the same neighbours, and, in the case of weighted graphs with possible loops, the corresponding incident edges have equal weights. In this paper, we explore the role of twin vertices in quantum state transfer. In particular, we provide characterizations of periodicity, perfect state transfer, and pretty good state transfer between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we provide characterizations of all simple unweighted double cones on regular graphs that exhibit periodicity, perfect state transfer, and pretty good state transfer. [ABSTRACT FROM AUTHOR]

We formalize the notion of a sedentary vertex and present a relaxation of the concept of a sedentary family of graphs introduced by Godsil (Linear Algebra Appl 614:356–375, 2021. https://doi.org/10.1016/j.laa.2020.08.027). We provide sufficient conditions for a given vertex in a graph to exhibit sedentariness. We also show that a vertex with at least two twins (vertices that share the same neighbours) is sedentary. We prove that there are infinitely many graphs containing strongly cospectral vertices that are sedentary, which reveals that, even though strong cospectrality is a necessary condition for pretty good state transfer, there are strongly cospectral vertices which resist high probability state transfer to other vertices. Moreover, we derive results about sedentariness in products of graphs which allow us to construct new sedentary families, such as Cartesian powers of complete graphs and stars. [ABSTRACT FROM AUTHOR]

In this article, we consider the relationships between walks in a signed graph G˙\dot{G} and its eigenvalues, with a particular focus on the largest absolute value of its eigenvalues ρ(G˙)\rho \left(\dot{G}), known as the spectral radius. Among other results, we derive a sequence of lower bounds for ρ(G˙)\rho \left(\dot{G}) expressed in terms of walks or closed walks. We also prove that ρ(G˙)\rho \left(\dot{G}) attains the spectral radius of the underlying graph GG if and only if G˙\dot{G} is switching equivalent to GG or its negation. It is proved that the length kk of the shortest negative cycle in G˙\dot{G} and the number of such cycles are determined by the spectrum of G˙\dot{G} and the spectrum of GG. Finally, a relation between kk and characteristic polynomials of G˙\dot{G} and GG is established.

We consider signed graphs with just 2 or 3 distinct eigenvalues, in particular (i) those with at least one simple eigenvalue, and (ii) those with vertex-deleted subgraphs which themselves have at most 3 distinct eigenvalues. We also construct new examples using weighing matrices and symmetric 3-class association schemes.

This article provides a combinatorial description of the inverse of the adjacency matrix of a non-singular 3-coloured digraph. The class of unicyclic 3-coloured digraphs with the cycle weight ±i and with a unique perfect matching, denoted by U g , is considered in this article. We characterize the 3-coloured digraphs in U g whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs in U g whose inverses are bipartite are also characterized. It is proved that the inverses of the 3-coloured digraphs in U g are always Laplacian non-singular. Characterizations of unicyclic 3-coloured digraphs in U g possessing unicyclic inverses are also supplied in this article. As an application, we can obtain the class of unicyclic 3-coloured digraphs with the cycle weight ±i satisfying the strong reciprocal eigenvalue property. [ABSTRACT FROM AUTHOR]

We investigate signed graphs with just 2 or 3 distinct eigenvalues, mostly in the context of vertex-deleted subgraphs, the join of two signed graphs or association schemes. [ABSTRACT FROM AUTHOR]

We study the balance of G-gain graphs, where G is an arbitrary group, by investigating their adjacency matrices and their spectra. As a first step, we characterize switching equivalence and balance of gain graphs in terms of their adjacency matrices in M n (C G) . Then we introduce a represented adjacency matrix, associated with a gain graph and a group representation, by extending the theory of Fourier transforms from the group algebra C G to the algebra M n (C G) . We prove that, anytime G admits a finite-dimensional faithful unitary representation π , a G-gain graph is balanced if and only if the spectrum of the represented adjacency matrix associated with π coincides with the spectrum of the underlying graph, with multiplicity given by the degree of the representation. We show that the complex adjacency matrix of complex unit gain graphs and the adjacency matrix of a cover graph are indeed particular cases of our construction. This enables us to recover some classical results and prove some new characterizations of balance in terms of spectrum, index or structure of these graphs. [ABSTRACT FROM AUTHOR]