Algebraic Connectivity for Vertex-Deleted Subgraphs, and a Notion of Vertex Centrality

Let G be a connected graph, suppose that v is a vertex of G, and denote the subgraph formed from G by deleting vertex v by G\v. Denote the algebraic connectivities of G and G\v by α(G) and (G\v), respectively. In this paper, we consider the functions ∅(v) = α(G)− α(G\v) and k(v) = α(G\v)/α(G), provide attainable upper and lower bounds on both functions, and characterise the equality cases in those bounds. The function yields a measure of vertex centrality, and we apply that measure to analyse certain graphs arising from food webs

On Q-spectral integral variation

Let G be a graph with two non adjacent vertices and G0 the graph constructed from G by adding an edge between them. It is known that the trace of Q0 is 2 plus the trace of Q, where Q and Q0 are the signless Laplacian matrices of G and G0 respectively. So, the sum of the Q0-eigenvalues of G0 is the sum of the the Q- eigenvalues of G plus two. It is said that Q-spectral integral variation occurs when either only one Q-eigenvalue is increased by two or two Q-eigenvalues are increased by 1 each one. In this article we present some conditions for the occurrence of Q-spectral integral variation under the addition of an edge to a graph G

A Cycle-Based Bound for Subdominant Eigenvalues of Stochastic Matrices

Given a primitive stochastic matrix, we provide an upper bound on the moduli of its non-Perron eigenvalues. The bound is given in terms of the weights of the cycles in the directed graph associated with the matrix. The bound is attainable in general, and we characterize a special case of equality when the stochastic matrix has a positive row. Applications to Leslie matrices and to Google-type matrices are also considere

On Q-spectral integral variation

Let G be a graph with two non adjacent vertices and G0 the graph constructed from G by adding an edge between them. It is known that the trace of Q0 is 2 plus the trace of Q, where Q and Q0 are the signless Laplacian matrices of G and G0 respectively. So, the sum of the Q0-eigenvalues of G0 is the sum of the the Q- eigenvalues of G plus two. It is said that Q-spectral integral variation occurs when either only one Q-eigenvalue is increased by two or two Q-eigenvalues are increased by 1 each one. In this article we present some conditions for the occurrence of Q-spectral integral variation under the addition of an edge to a graph G

A Cycle-Based Bound for Subdominant Eigenvalues of Stochastic Matrices

Given a primitive stochastic matrix, we provide an upper bound on the moduli of its non-Perron eigenvalues. The bound is given in terms of the weights of the cycles in the directed graph associated with the matrix. The bound is attainable in general, and we characterize a special case of equality when the stochastic matrix has a positive row. Applications to Leslie matrices and to Google-type matrices are also considere

The Markov chain tree theorem and the state reduction algorithm in commutative semirings

We extend the Markov chain tree theorem to general commutative semirings, and we generalize the state reduction algorithm to commutative semifields. This leads to a new universal algorithm, whose prototype is the state reduction algorithm which computes the Markov chain tree vector of a stochastic matrix.Comment: 13 page

Quantum walks on join graphs

The join $X\vee Y$ of two graphs $X$ and $Y$ is the graph obtained by joining each vertex of $X$ to each vertex of $Y$. We explore the behaviour of a continuous quantum walk on a weighted join graph having the adjacency matrix or Laplacian matrix as its associated Hamiltonian. We characterize strong cospectrality, periodicity and perfect state transfer (PST) in a join graph. We also determine conditions in which strong cospectrality, periodicity and PST are preserved in the join. Under certain conditions, we show that there are graphs with no PST that exhibits PST when joined by another graph. This suggests that the join operation is promising in producing new graphs with PST. Moreover, for a periodic vertex in $X$ and $X\vee Y$, we give an expression that relates its minimum periods in $X$ and $X\vee Y$. While the join operation need not preserve periodicity and PST, we show that $\big| |U_M(X\vee Y,t)_{u,v}|-|U_M(X,t)_{u,v}| \big|\leq \frac{2}{|V(X)|}$ for all vertices $u$ and $v$ of $X$, where $U_M(X\vee Y,t)$ and $U_M(X,t)$ denote the transition matrices of $X\vee Y$ and $X$ respectively relative to either the adjacency or Laplacian matrix. We demonstrate that the bound $\frac{2}{|V(X)|}$ is tight for infinite families of graphs.Comment: 29 page