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A complex unit gain graph (or T{\mathbb{T}}-gain graph) Γ=(G,γ)\Gamma =\left(G,\gamma ) is a gain graph with gains in T{\mathbb{T}}, the multiplicative group of complex units. The T{\mathbb{T}}-outgain in Γ\Gamma of a vertex v∈Gv\in G is the sum of the gains of all the arcs originating in vv. A T{\mathbb{T}}-gain graph is said to be an aa-T{\mathbb{T}}-regular graph if the T{\mathbb{T}}-outgain of each of its vertices is equal to aa. In this article, it is proved that aa-T{\mathbb{T}}-regular graphs exist for every a∈Ra\in {\mathbb{R}}. This, in particular, means that every real number can be a T{\mathbb{T}}-gain graph eigenvalue. Moreover, denoted by Ω(a)\Omega \left(a) the class of connected T{\mathbb{T}}-gain graphs whose largest eigenvalue is the real number aa, it is shown that Ω(a)\Omega \left(a) is nonempty if and only if aa belongs to {0}∪[1,+∞)\left\{0\right\}\cup \left[1,+\infty ). In order to achieve these results, non-complete extended pp-sums and suitably defined joins of T{\mathbb{T}}-gain graphs are considered.

Let 𝕋4 = {±1, ±i} be the subgroup of 4-th roots of unity inside 𝕋, the multiplicative group of complex units. A complex unit gain graph Φ is a simple graph Γ = (V (Γ) = {v1, . . . , vn}, E(Γ)) equipped with a map ϕ:E→(Γ)→𝕋\varphi :\vec E(\Gamma ) \to \mathbb{T} defined on the set of oriented edges such that ϕ(vivj) = ϕ(vjvi)−1. The gain graph Φ is said to be balanced if for every cycle C = vi1vi2vikvi1 we have ϕ(vi1vi2)ϕ(vi2vi3) ϕ(vikvi1) = 1.

Let G be a graph with order n and size m. For each real number α ∈ [ 0 , 1 ] , the A α -matrix of a graph G is defined as A α (G) := α D (G) + (1 − α) A (G) , where D (G) = d i a g (d 1 , ... , d n) is the degree matrix of G, and A (G) is its adjacency matrix. The functions s (G) = ∑ 1 ≤ i ≤ n d i − 2 m n a n d v a r (G) = 1 n ∑ 1 ≤ i ≤ n d i − 2 m n 2 have been already used elsewhere as measures of graph irregularity. In this paper, we find several bounds for the spectral radius λ (G) of the matrix A α (G). Especially, when G is non-empty we prove for α ∈ [ 0 , 1) that s 2 (G) 2 n 2 n (2 α − α 2) Δ 2 + 2 (1 − α) 2 m ≤ λ (G) − 2 m n ≤ 1 + (2 α − α 2) n Ω (G) 2 + (1 − α) 2 s (G) , where Ω (G) = Δ (G) − ⌈ 2 m / n ⌉ + 1 and Δ (G) is the maximum vertex degree of G. These inequalities extend to the A α -matrix a Nikiforov's result involving the spectral radius of the adjacency matrix. In addition, for fixed order and size, we determine the graphs minimizing the degree variance v a r (G) and compute the A α -spectrum of some blown-up graphs. [ABSTRACT FROM AUTHOR]

A well-known fact in Spectral Graph Theory is the existence of pairs of cospectral (or isospectral) nonisomorphic graphs, known as PINGS. The work of A.J. Schwenk (in 1973) and of C. Godsil and B. McKay (in 1982) shed some light on the explanation of the presence of cospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil–McKay-type routines developed for graphs, whose adjacency matrices are (0 , 1) -matrices, to the level of signed graphs, whose adjacency matrices allow the presence of − 1 s. We show that, with suitable adaption, such routines can be successfully ported to signed graphs, and we can build pairs of cospectral switching nonisomorphic signed graphs. [ABSTRACT FROM AUTHOR]

A signed graph is a pair Γ = (G , σ) , where G = (V (G) , E (G)) is a graph and σ : E (G) → { + 1 , − 1 } is the sign function on the edges of G. For a signed graph we consider the Laplacian matrix defined as L (Γ) = D (G) − A (Γ) , where D (G) is the matrix of vertex degrees of G and the (signed) adjacency matrix A (Γ). A signed ∞-graph consists of two signed cycles with just one vertex in common. In this paper, we study the Laplacian spectral determination problem for the class of signed ∞-graphs, and we identify all connected L-cospectral mates. [ABSTRACT FROM AUTHOR]

A signed graph is a pair Γ = ( G , σ ) , where G = ( V ( G ) , E ( G ) ) is a graph and σ : E ( G ) → { + 1 , − 1 } is the sign function on the edges of G . For a signed graph we consider the Laplacian matrix defined as L ( Γ ) = D ( G ) − A ( Γ ) , where D ( G ) is the matrix of vertices degrees of G and A ( Γ ) is the (signed) adjacency matrix. The least Laplacian eigenvalue is zero if and only if the signed graph is balanced, i.e. all cycles contain an even number of negative edges. Here we show that among the unbalanced bicyclic signed graphs of given order n ≥ 5 the least Laplacian eigenvalue is minimal for signed graphs consisting of two triangles, only one of which is unbalanced, connected by a path. We also identify the signed graphs minimizing the least eigenvalue among those whose unbalanced (bicyclic) base is a theta-graph. [ABSTRACT FROM AUTHOR]