The corona G◦H of two graphs G and H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining each i-th vertex of G to every vertex in the i-th copy of H. The neighborhood coronaG⋆H of two graphs G and H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the neighbors of the i-th vertex of G to every vertex in the i-th copy of H. In this p...

Let G be a graph with adjacency matrix A(G) and let D(G) the diagonal of degrees G. For every real α ∈ [0, 1], Nikiforov [21] Wang et al. [26] defined matrices Aα(G) Lα(G), respectively, as = αD(G)+(1−α)A(G) Lα(G) αD(G)+(α − 1)A(G). In this paper, we obtain some relationships between eigenvalues these for families graphs, part Aα Lα-spectrum spider display Lα-characteristic polynomials when the...

Let G be a simple, connected graph and let A(G) the adjacency matrix of G. If D(G) is diagonal vertex degrees G, then for every real $$\alpha \in [0,1]$$ , $$A_{\alpha }(G)$$ defined as }(G) = \alpha + (1- ) A(G).$$ The eigenvalues form }$$ -spectrum $$G_1 {\dot{\vee }} G_2$$ {\underline{\vee \langle \text {v} \rangle {e} denote subdivision-vertex join, subdivision-edge R-vertex join R-edge two...

In order to obtain perfect state transfer between two sites in a network of interacting qubits, their corresponding vertices in the underlying graph must satisfy a property called strong cospectrality. Here we determine the structure of graphs containing pairs of vertices which are strongly cospectral and satisfy a certain extremal property related to the spectrum of the graph. If the graph sat...

We show that a number of graph invariants are, even combined, insufficient to distinguish between nonisomorphic trees or general graphs. Among these are: the spectrum of eigenvalues (equivalently, the characteristic polynomial), the number of independent sets of all sizes or the number of connected subgraphs of all sizes. We therefore extend the classical theorem of Schwenk that almost every tr...

A complex unit gain graph ( T -gain graph) Φ = G , φ ) is a where the function assigns number to each orientation of an edge and its inverse assigned opposite orientation. balanced if product gains oriented cycle (if any) 1. We propose two notions distance matrices D < max min Φ, for any ordering ‘<’ vertex set. characterize graphs which are independent ordering. show holds standard vertices on...

Given a graph $G$, the exponential distance matrix is defined entry-wise by letting $(u,v)$-entry be $q^{\text{dist}(u,v)}$, where $\text{dist}(u,v)$ between vertices $u$ and $v$ with convention that if are in different components, then $q^{\text{dist}(u,v)}=0$. In this paper, we will establish several properties of characteristic polynomial (spectrum) for matrix, give some families graphs whic...