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A complex unit gain graph ($ \mathbb{T} $-gain graph), $ \Phi=(G, \varphi) $ is a graph where the function $ \varphi $ assigns a unit complex number to each orientation of an edge of $ G $, and its inverse is assigned to the opposite orientation. %A complex unit gain graph($ \mathbb{T} $-gain graph) is a simple graph where each orientation of an edge is given a complex unit, and its inverse is assigned to the opposite orientation of the edge. In this article, we propose gain distance matrices for $ \mathbb{T} $-gain graphs. These notions generalize the corresponding known concepts of distance matrices and signed distance matrices. Shahul K. Hameed et al. introduced signed distance matrices and developed their properties. Motivated by their work, we establish several spectral properties, including some equivalences between balanced $ \mathbb{T} $-gain graphs and gain distance matrices. Furthermore, we introduce the notion of positively weighted $ \mathbb{T} $-gain graphs and study some of their properties. Using these properties, Acharya's and Stani\'c's spectral criteria for balance are deduced. Moreover, the notions of order independence and distance compatibility are studied. Besides, we obtain some characterizations for distance compatibility., Comment: 20 pages; 2 figures

Let $ \Phi=(G, \varphi) $ be a connected complex unit gain graph ($ \mathbb{T} $-gain graph) on a simple graph $ G $ with $ n $ vertices and maximum vertex degree $ \Delta $. The associated adjacency matrix and degree matrix are denoted by $ A(\Phi) $ and $ D(\Phi) $, respectively. Let $ m_{\alpha}(\Phi,\lambda) $ be the multiplicity of $ \lambda $ as an eigenvalue of $ A_{\alpha}(\Phi) :=\alpha D(\Phi)+(1-\alpha)A(\Phi)$, for $ \alpha\in[0,1) $. In this article, we establish that $ m_{\alpha}(\Phi, \lambda)\leq \frac{(\Delta-2)n+2}{\Delta-1}$, and characterize the classes of graphs for which the equality hold. Furthermore, we establish a couple of bounds for the rank of $A(\Phi)$ in terms of the maximum vertex degree and the number of vertices. One of the main results extends a result known for unweighted graphs and simplifies the proof in [15], and other results provide better bounds for $r(\Phi)$ than the bounds known in [8].

Let $ ��=(G, ��) $ be a connected complex unit gain graph ($ \mathbb{T} $-gain graph) on a simple graph $ G $ with $ n $ vertices and maximum vertex degree $ ��$. The associated adjacency matrix and degree matrix are denoted by $ A(��) $ and $ D(��) $, respectively. Let $ m_��(��,��) $ be the multiplicity of $ ��$ as an eigenvalue of $ A_��(��) :=��D(��)+(1-��)A(��)$, for $ ��\in[0,1) $. In this article, we establish that $ m_��(��, ��)\leq \frac{(��-2)n+2}{��-1}$, and characterize the classes of graphs for which the equality hold. Furthermore, we establish a couple of bounds for the rank of $A(��)$ in terms of the maximum vertex degree and the number of vertices. One of the main results extends a result known for unweighted graphs and simplifies the proof in [15], and other results provide better bounds for $r(��)$ than the bounds known in [8].

A $\mathbb{T}$-gain graph, $\Phi = (G, \varphi)$, is a graph in which the function $\varphi$ assigns a unit complex number to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix $ A(\Phi) $ is defined canonically. The energy $ \mathcal{E}(\Phi) $ of a $ \mathbb{T} $-gain graph $ \Phi $ is the sum of the absolute values of all eigenvalues of $ A(\Phi) $. We study the notion of energy of a vertex of a $ \mathbb{T} $-gain graph, and establish bounds for it. For any $ \mathbb{T} $-gain graph $ \Phi$, we prove that $2\tau(G)-2c(G) \leq \mathcal{E}(\Phi) \leq 2\tau(G)\sqrt{\Delta(G)}$, where $ \tau(G), c(G)$ and $ \Delta(G)$ are the vertex cover number, the number of odd cycles and the largest vertex degree of $ G $, respectively. Furthermore, using the properties of vertex energy, we characterize the classes of $ \mathbb{T} $-gain graphs for which $ \mathcal{E}(\Phi)=2\tau(G)-2c(G) $ holds. Also, we characterize the classes of $ \mathbb{T} $-gain graphs for which $\mathcal{E}(\Phi)= 2\tau(G)\sqrt{\Delta(G)} $ holds. This characterization solves a general version of an open problem. In addition, we establish bounds for the energy in terms of the spectral radius of the associated adjacency matrix., Comment: 31 pages, 4 figures

A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is called $\mathbb{T}$-gain adjacency matrix. Let $\mathbb{T}_{G} $ denote the collection of all $\mathbb{T}$-gain adjacency matrices on a graph $G$. In this article, we study the cospectrality of matrices in $\mathbb{T}_{G} $ and we establish equivalent conditions for a graph $G$ to be a tree in terms of the spectrum and the spectral radius of matrices in $\mathbb{T}_{G} $. We identify a class of connected graphs $\mathfrak{F^{'}}$ such that for each $G \in \mathfrak{F^{'}}$, the matrices in $\mathbb{T}_G$ have nonnegative real part up to diagonal unitary similarity. Then we establish bounds for the spectral radius of $\mathbb{T}$-gain adjacency matrices on $ G \in \mathfrak{F^{'}} $ in terms of their largest eigenvalues. Thereupon, we characterize $\mathbb{T}$-gain graphs for which the spectral radius of the associated $\mathbb{T}$-gain adjacency matrices equal to the largest vertex degree of the underlying graph. These bounds generalize results known for the spectral radius of Hermitian adjacency matrices of digraphs and provide an alternate proof of a result about the sharpness of the bound in terms of largest vertex degree established in [Krystal Guo, Bojan Mohar. Hermitian adjacency matrix of digraphs and mixed graphs. J. Graph Theory 85 (2017), no. 1, 217-248.]., Comment: Comments are welcome!