Search

A mixed graph M G is the graph obtained from an unoriented simple graph G by giving directions to some edges of G, where G is often called the underlying graph of M G . In this paper, we discuss the determinants of two classes of incidence matrices of the second kind of rootless mixed trees and unicyclic mixed graphs. Applying these results, we characterize the explicit expressions of various minors for Hermitian (quasi-)Laplacian matrix of the second kind of M G . Moreover, we give two sufficient conditions that the module of any cofactor of Hermitian (quasi-)Laplacian matrix of the second kind of the mixed graph M G is equal to the number of spanning trees of the underlying graph G. [ABSTRACT FROM AUTHOR]

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. Recently, there has been a renewed interest in studying Seidel spectrum of graphs and some achievements have been made in this regard. In this paper, we determine the Seidel spectra of some variants of corona operations, such as edge corona, subdivision-vertex corona, subdivision-vertex neighbourhood corona of two graphs and so on. The corresponding Seidel eigenvectors are also described completely. Applying the obtained results, we give some sufficient and necessary conditions for edge corona, subdivision-vertex corona and subdivision-vertex neighbourhood corona of two graphs to be Seidel integral. [ABSTRACT FROM AUTHOR]

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. For two simple connected graphs G 1 and G 2 , let us denote the subdivision-vertex join and subdivision-edge join by G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 , respectively. In this paper, we completely determine the Seidel spectrum and corresponding Seidel eigenvectors of G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 . As an application, we give a sufficient and necessary condition for G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 to be Seidel integral. [ABSTRACT FROM AUTHOR]

Given two graphs G 1 of order n 1 and G 2 , the neighborhood corona of G 1 and G 2 , denoted by G 1 ★ G 2 , is the graph obtained by taking one copy of G 1 and taking n 1 copies of G 2 , in the meanwhile, linking all the neighbors of the i-th vertex of G 1 with all vertices of the i-th copy of G 2 . In our work, we give some conditions that G 1 ★ G 2 is not periodic. Furthermore, we demonstrate some sufficient conditions for G 1 ★ G 2 having no perfect state transfer. Some examples are provided to explain our results. In addition, for the reason that the graph admitting perfect state transfer is rare, we also consider pretty good state transfer on neighborhood corona of two graphs. We show some sufficient conditions for G 1 ★ G 2 admitting pretty good state transfer. [ABSTRACT FROM AUTHOR]

Let S(G) be the Seidel matrix of a graph G. The Seidel energy of G, denoted by ES(G), is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix S(G) of G. In this paper, we consider the change of Seidel energy of graphs due to edge deletion. It is proved that the Seidel energy of the tripartite Turán graph T(n, 3) of order n ≥ 9 is always increased when an edge is deleted. [ABSTRACT FROM AUTHOR]

A graph G on n vertices is called non-universal if its maximum degree is at most n - 2 . In this paper, we give a structural characterization for non-universal maximal planar graphs with diameter two. In precise, we find 10 basic graphs, and then generate all 25 non-universal maximal planar graphs with diameter two by adding repeatedly and appropriately 3-vertices to some of these 10 basic graphs. As an application, we show that maximal planar graphs with diameter two are pancyclic except five special graphs. [ABSTRACT FROM AUTHOR]

For a simple connected graph G, the Q-generating function of the numbers N k of semi-edge walks of length k in G is defined by W Q (t) = ∑ k = 0 ∞ N k t k . This paper reveals that the Q-generating function W Q (t) may be expressed in terms of the Q-polynomials of the graph G and its complement G ¯ . Using this result, we study some Q-spectral properties of graphs and compute the Q-polynomials for some graphs obtained from various graph operations, such as the complement graph of a regular graph, the join of two graphs and the (edge)corona of two graphs. As another application of the Q-generating function W Q (t) , we also give a combinatorial interpretation of the Q-coronal of G, which is defined to be the sum of the entries of the matrix (λ I n - Q (G)) - 1 . This result may be used to obtain the many alternative calculations of the Q-polynomials of the (edge)corona of two graphs. Further, we also compute the Q-generating functions of the join of two graphs and the complete multipartite graphs. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with vertex set V (G) = { v 1 , v 2 , ... , v n } and edge set E (G). The arithmetic–geometric matrix A AG (G) = (h i j) of G is a matrix of order n defined by h i , j = 1 2 d i d j + d j d i if v i v j ∈ E (G) and 0 otherwise, where d i stands for the degree of the vertex v i in G. The arithmetic–geometric characteristic polynomial of G is the characteristic polynomial of A AG (G). The arithmetic–geometric energy ℰ AG (G) of G is the sum of absolute values of all eigenvalues of A AG (G). In this paper, we obtain the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy of some specific graphs. In addition, we also consider the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy change of these graphs when one edge is deleted. [ABSTRACT FROM AUTHOR]

For two graphs G and H, the corona product G ∘ H is the graph obtained by taking one copy of G and | V G | copies of H, and joining the ith vertex of G with every vertex of the ith copy of H. In this paper, we study the state transfer of corona relative to the signless Laplacian matrix. We explore some conditions that guarantee the signless Laplacian perfect state transfer in G ∘ H . We prove that G ∘ K m has no signless Laplacian perfect state transfer for some special m. We also show that K 2 ∘ H has pretty good state transfer but no perfect state transfer relative to the signless Laplacian matrix for a regular graph H. Furthermore, we show that n K 2 ¯ ∘ K 1 has signless Laplacian pretty good state transfer, where n K 2 ¯ is the cocktail party graph. [ABSTRACT FROM AUTHOR]

Many variants of join operations of graphs have been introduced, and their spectral properties have been studied extensively by many researchers. This paper mainly focuses on the Laplacian spectra of some double join operations of graphs. We first introduce the conception of double join matrix and provide a complete information about its eigenvalues and the corresponding eigenvectors. Further, we define four variants of double join operations based on subdivision graph, Q-graph, R-graph and total graph. Applying the result obtained about double join matrices, we give an explicit complete characterization of the Laplacian eigenvalues and the corresponding eigenvectors of four variants in terms of the Laplacian eigenvalues and the eigenvectors of factor graphs. These results generalize some well-known results on the Laplacian spectra of some join operations of graphs. [ABSTRACT FROM AUTHOR]

We establish an inclusion relation between two known inclusion intervals of matrix singular values in some special case. In addition, based on the use of positive scale vectors, a known inclusion interval of matrix singular values is also improved. [ABSTRACT FROM AUTHOR]