Jul 2026· The Electronic Journal of Linear Algebra· Vol 42, pp. 551-568· 0 citations
Abstract
A threshold graph is generated from a single node by repeatedly adding either a node $i$ connected to all existing nodes with a common link weight $w_i >0 $ or a node $i$ connected to none. Let $ G_w $ be a weighted threshold graph encoded by the weight vector $ w = (w_1, w_2, \ldots, w_N) $ with $w_i \geq 0$. A closed-form expression for the pseudoinverse of its Laplacian matrix $Q_w$ is derived via spectral decomposition, which yields an explicit formula for the effective resistance matrix $ \Omega_w $. We present a detailed structural characterization of the matrix $\Omega_w$ and determine a subset of the spectrum of the matrix $ \Omega_w $ in terms of the weights $w_i$. As an application, we show that when the missing links of a threshold graph are sequentially added in nondecreasing order of effective resistance, the threshold property of the graph is preserved at each step until the complete graph of the same size is obtained.
Each graphon W:Ω2→[0,1]$$ W:{\Omega}^2\to \left[0,1\right] $$ yields an inhomogeneous random graph model 𝔾(n,W) . We show that 𝔾(n,W) is asymptotically almost surely connected if and only if (i) W$$ W $$ is a connected graphon and (ii) the measure of elements of Ω$$ \Omega $$ of W$$ W $$ ‐degree less than α$$ \alpha $$...
J. Hladký, Gopal Viswanathan· Random Structures & Algo...· 1 citation· ⚡1
Let $G$ be a graph of order $n$ with adjacency matrix $A_G$, let $\mathbf e$ denote the all-one vector, and let$W_G=[\mathbf e,A_G\mathbf e,\ldots,A_G^{n-1}\mathbf e]$ be its walk matrix. We consider the case $\operatorname{rank}W_G=n-2$, the first corank for which distinct graphs can have the same walk matrix. We give...
For any graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A \in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ represent the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we p...
Jane Breen, Shaun M. Fallat, Johnna Parenteau· 0 citations
For a graph $G$, let $f_t(G)$ denote the minimum of the maximum degree of an induced subgraph with $\alpha(G)+t$ vertices, where $\alpha(G)$ is the independence number, and write $f(G)=f_1(G)$. Huang's theorem gives $f(Q_k)\ge\lceil\sqrt{k}\rceil$ for the $k$-dimensional hypercube $Q_k$. We extend this lower bound to C...
Let $G$ be a graph of order $n$. The hamming matrix $H(G) = [h_{ij}]$ of $G$ is an $n \times n$ matrix whose $(i,j)$-entry is the hamming distance between the strings $s(v_i)$ and $s(v_j)$. The hamming energy $HE(G)$ of a graph $G$ is the sum of the absolute values of the eigenvalues of $H(G)$. In this paper, we study...
K. P., M. A. Sriraj, S. V. Roopa· Boletim da Sociedade Paranae...· 0 citations
Let $G$ be a connected graph, and let $\lambda_1(G)>\lambda_2(G)$ denote its two largest adjacency eigenvalues. The spectral gap of $G$ is defined as the difference $\lambda_1(G) - \lambda_2(G)$. For integers $r\geq 2$ and $s\geq 0$, the double kite $DK(r,s)$ is formed by taking two vertex-disjoint copies of the comple...
Le-Le Liu, Michael Tait, Yi Wang· 0 citations
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