Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHermitian adjacency matrix of the second kind for mixed graphs
254
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind introduced by Mohar [21]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from $u$ to $v$ is equal to the sixth root of unity $omega=frac{1+{bf i}sqrt{3}}{2}$ (and its symmetric entry is $overline{omega}=frac{1-{bf i}sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: Some interesting properties are discovered about the characteristic polynomial of this novel matrix. Cospectral problems among mixed graphs, including mixed graphs and their underlying graphs, are studied. We give equivalent conditions for a mixed graph that shares the same spectrum of its Hermitian adjacency matrix of the second kind ($H_S$-spectrum for short) with its underlying graph. A sharp upper bound on the $H_S$-spectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called three-way switchings are discussed--they give rise to a large number of $H_S$-cospectral mixed graphs. We extract all the mixed graphs whose rank of its Hermitian adjacency matrix of the second kind ($H_S$-rank for short) is $2$ (resp. 3). Furthermore, we show that all connected mixed graphs with $H_S$-rank $2$ can be determined by their $H_S$-spectrum. However, this does not hold for all connected mixed graphs with $H_S$-rank $3$. We identify all mixed graphs whose eigenvalues of its Hermitian adjacency matrix of the second kind ($H_S$-eigenvalues for short) lie in the range $(-alpha,, alpha)$ for $alphainleft{sqrt{2},,sqrt{3},,2right}$.
rate research
Read More
In 1960, Hoffman and Singleton cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$ (kappa - 1) I_n + J_n - A A^{rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all one matrix, and a $(0,1)$--matrix with all row and column sums equal to $kappa$, respectively. If $A$ is an incidence matrix of some configuration $cal C$ of type $n_kappa$, then the left-hand side $Theta(A):= (kappa - 1)I_n + J_n - A A^{rm T}$ is an adjacency matrix of the non--collinearity graph $Gamma$ of $cal C$. In certain situations, $Theta(A)$ is also an incidence matrix of some $n_kappa$ configuration, namely the neighbourhood geometry of $Gamma$ introduced by Lef`evre-Percsy, Percsy, and Leemans cite{LPPL}. The matrix operator $Theta$ can be reiterated and we pose the problem of solving the generalised Hoffman--Singleton equation $Theta^m(A)=A$. In particular, we classify all $(0,1)$--matrices $M$ with all row and column sums equal to $kappa$, for $kappa = 3,4$, which are solutions of this equation. As a by--product, we obtain characterisations for incidence matrices of the configuration $10_3F$ in Kantors list cite{Kantor} and the $17_4$ configuration $#1971$ in Betten and Bettens list cite{BB99}.
During routine state space circuit analysis of an arbitrarily connected set of nodes representing a lossless LC network, a matrix was formed that was observed to implicitly capture connectivity of the nodes in a graph similar to the conventional incidence matrix, but in a slightly different manner. This matrix has only 0, 1 or -1 as its elements. A sense of direction (of the graph formed by the nodes) is inherently encoded in the matrix because of the presence of -1. It differs from the incidence matrix because of leaving out the datum node from the matrix. Calling this matrix as forward adjacency matrix, it was found that its inverse also displays useful and interesting physical properties when a specific style of node-indexing is adopted for the nodes in the graph. The graph considered is connected but does not have any closed loop/cycle (corresponding to closed loop of inductors in a circuit) as with its presence the matrix is not invertible. Incidentally, by definition the graph being considered is a tree. The properties of the forward adjacency matrix and its inverse, along with rigorous proof, are presented.
It is well known that spectral Tur{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur{a}n type problem. Let $G$ be a graph and let $mathcal{G}$ be a set of graphs, we say $G$ is textit{$mathcal{G}$-free} if $G$ does not contain any element of $mathcal{G}$ as a subgraph. Denote by $lambda_1$ and $lambda_2$ the largest and the second largest eigenvalues of the adjacency matrix $A(G)$ of $G,$ respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on $lambda_1^{2k}+lambda_2^{2k}$ of $n$-vertex ${C_3,C_5,ldots,C_{2k+1}}$-free graphs is established, where $k$ is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most $2k+1$ in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of $n$-vertex non-bipartite graphs with odd girth at least $2k+3,$ which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].
We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix, and clarify explicit formulas of their supports of the square. In addition, we give tables by computer on the identification of digraphs by their eigenvalues.
The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously is a relevant problem in data driven modeling of networks. Here we formulate common generalizations of this problem and the Exact Matching Problem, and solve them with an algebraic Monte-Carlo algorithm that runs in polynomial time if the number of partition classes is bounded.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
