The net Laplacian matrix of a signed graph \(\Gamma = (G, \sigma)\), where \(G = (V(G),E(G))\) is an unsigned graph (referred to as the underlying graph) and \(\sigma: E(G) \rightarrow \{-1, +1\}\) is the sign function, is defined as \(L^{\pm}(\Gamma) = D^{\pm}(\Gamma) - A(\Gamma)\). Here, \(D^{\pm}(\Gamma)\) and \(A(\Gamma)\) represent the diagonal matrix of net-degrees and the adjacency matrix of \(\Gamma\), respectively. The nullity of \(L^{\pm}(\Gamma)\), denoted as \(\eta (L^{\pm} (\Gamma))\), refers to the multiplicity of \(0\) as an eigenvalue of \(L^{\pm}(\Gamma)\). In this paper, we concentrate on the nullity of the net Laplacian matrix of a connected signed graph \(\Gamma\), and establish that \(1 \leq \eta (L^{\pm} (\Gamma)) \leq \min\{ \beta(\Gamma) + 1, |V(\Gamma)| - 1 \}\), where \(\beta(\Gamma) = |E(\Gamma)| - |V(\Gamma)| + 1\) denotes the cyclomatic number of \(\Gamma\). We completely determine the connected signed graphs with nullity \(|V(\Gamma)| - 1\). Additionally, we characterize the signed cactus graphs with nullity \(1\) or \(\beta(\Gamma) + 1\).

The Four point condition (abbreviated as 4PC) is a condition used to test if a given distance matrix arises from shortest path distances on trees. From a tree \(T\), Bapat and Sivasubramanian defined a matrix \(\operatorname{Min4PC}_T\) based on this condition. They also gave a basis \(B\) for the row space of \(\operatorname{Min4PC}_T\) and determined its Smith Normal Form. In this paper, we consider the matrix \(\operatorname{Min4PC}_T[B,B]\) restricted to a basis \(B\) and give an explicit inverse for it. It is known that the distance matrix \(D_T\) of a tree \(T\), is invertible and that its inverse is a rank-one update of its scaled Laplacian. Our inverse has a similar form and is a rank-one update of a Laplacian like matrix.

Let \(G\) be a connected graph. A matching \(M\) in \(G\) is a set of edges of \(G\) without two of them adjacent (having a common vertex). The graph whose vertices are the matchings in \(G\) and two matchings \(M\) and \(N\) are adjacent if and only if \( (M\setminus N)\cup (N\setminus M)\) is the edge set of a path or a cycle, is denoted by \({\cal G}({\cal M}(G))\) and called the skeleton of the matching polytope of \(G\). The degree of a matching \(M\) in \(G\) is the degree of the vertex \(M\) in \({\cal G}({\cal M}(G))\). In the literature some authors have studied the degree of some matchings, in particular when \(G\) is a tree. In this paper we continue this study and we present some formulas to compute the degree of a matching \(M\) in a graph with cycles. More explicitly, we focus on the matchings having \(1\) or \(2\) edges and on the matchings of more than two edges with some constraints.

Let \(G\) be a bipartite graph with adjacency matrix \(A(G)\). The characteristic polynomial \(\phi(G,x)=\det(xI-A(G))\) and the permanental polynomial \(\pi(G,x) = \operatorname{per}(xI-A(G))\) are both graph invariants used to distinguish graphs. For bipartite graphs, we define the modified characteristic polynomial, which is obtained by changing the signs of some of the coefficients of \(\phi(G,x)\). For \(4k\)-intercyclic bipartite graphs, i.e., those for which the removal of any \(4k\)-cycle results in a \(C_{4k}\)-free graph, we provide an expression for \(\pi(G,x)\) in terms of the modified characteristic polynomial of the graph and its subgraphs. Our approach is purely combinatorial in contrast to the Pfaffian orientation method found in the literature to compute the permanental polynomial.

An induced subgraph of a graph \(G\) is said to be dominating if every vertex of \(G\) is at distance at most one from this subgraph. We investigate pairs \((G, F)\) where \(F\) is a non-singular dominating induced subgraph of \(G,\) and the rank of \(G\) (that is, the rank of its adjacency matrix) attains the minimum, i.e., equals the number of vertices in \(F\). It turns out that the inverse of the adjacency matrix of a non-singular path, half graph, or even cycle is the adjacency matrix of a related signed graph; here, a half graph refers to a connected chain graph with exactly one vertex in each cell. We exploit this property to give a complete characterization of graphs \(G\) paired with any of these graphs in the role of \(F\). The bipartite case is singled out. It occurs that every nonsingular \(F\) is paired with an infinite family of graphs \(G\), and their number is comparatively large even if we exclude the existence of the so-called twin vertices. The latter empirical observation is demonstrated through some examples.

It has been an open problem to find the Moore-Penrose inverses of the incidence, Laplacian, and signless Laplacian matrices of families of graphs except trees and unicyclic graphs. Since the inverse formulas for an odd unicyclic graph and an even unicyclic graph are quite different, we consider wheel graphs as they are formed from odd or even cycles. In this article we solve the open problem for wheel graphs. This work has an interesting connection to inverses of circulant matrices.

In this article we present an infinite family of bipartite distance biregular graphs having an arbitrarily large diameter and whose distance matrices have exactly four distinct eigenvalues. This result answers a question posed by F. Atik and P. Panigrahi in On the distance spectrum of distance regular graphs (Linear Algebra and its Applications, 478 (2015), pp. 256-273) about the existence of connected graphs with diameter \(d\) that are not distance regular, whose distance matrix has less than \(d + 1\) distinct eigenvalues.

A matrix is called acyclic if replacing the diagonal entries with \(0\), and the nonzero diagonal entries with \(1\), yields the adjacency matrix of a forest. In this paper we show that the null space and the rank of an acyclic matrix with \(0\) in the diagonal is obtained from the null space and the rank of the adjacency matrix of the forest by multipliying by nonsingular diagonal matrices. We combine these with an algorithm for finding a sparsest basis of the null space of a forest to provide an optimal time algorithm for finding a sparsest basis of the null space of acyclic matrices with \(0\) in the diagonal.

A square matrix is called a multipart matrix if all its diagonal entries are zero and all other entries in each column are constant. In this paper, we describe various interesting spectral properties of multipart matrices. We provide suitable bounds for the spectral radius of a multipart matrix. Later on, we show applications of multipart matrices in spectral graph theory.

A signed cactus \(\dot{G}\) is a connected signed graph such that every edge belongs to at most one cycle. The rank of \(\dot{G}\) is the rank of its adjacency matrix. In this paper we prove that \[\sum_{i=1}^k n_i-2k\leq \operatorname{rank}(\dot{G})\leq \sum_{i=1}^k n_i-2t +2 s,\] where \(k\) is the number of cycles in \(\dot{G}\), \(n_1, n_2, \ldots, n_k\) are their lengths, \(t\) is the number of cycles whose rank is their order minus two, and \(s\) is the number of edges outside cycles. Signed cacti attaining the lower bound are determined.

We address the problem of ordering trees with the same degree sequence by their spectral radii. To achieve that, we consider 2-switch transformations which preserve the degree sequence and establish when the index decreases. Our main contribution is to determine a total ordering of a particular family by their indices according to a given parameter related to sizes in the tree.

We propose the notion of normalized Laplacian matrix \(\mathcal{L}(\Phi)\) for a gain graph \(\Phi\) and study its properties in detail, providing insights and counterexamples along the way. We establish bounds for the eigenvalues of \(\mathcal{L}(\Phi)\) and characterize the classes of graphs for which equality holds. The relationships between the balancedness, bipartiteness, and their connection to the spectrum of \(\mathcal{L}(\Phi)\) are also studied. Besides, we extend the edge version of eigenvalue interlacing for the gain graphs. Thereupon, we determine the coefficients for the characteristic polynomial of \(\mathcal{L}(\Phi)\).

The union-closed sets conjecture states that if a finite set \(\mathcal A\) of finite sets is union-closed and \(\mathcal A\neq \{ \varnothing\}\), then there exists an element in \(\displaystyle\cup_{A\in \mathcal A} A\) that belongs to at least half of the sets in \(\mathcal A\). We present three new formulations of the union-closed conjecture in terms of matrices, graphs, and hypergraphs.

In this paper, we compute the Gallai-Edmonds decomposition of a unicyclic graph \(G\) using linear algebraic tools. More precisely, the Gallai-Edmonds decomposition of \(G\) is obtained from the null space associated with adjacency matrices of its subtrees.

Let \(\dot{G}\) be a signed graph with \(n\) vertices and the frustration index \(\ell\). We prove the existence of \(k~(k\geq \ell)\) edges \(e_1, e_2, \ldots, e_k\) of \(\dot{G}\) such that \[\begin{equation*}\lambda_1(\dot{G})\leq \lambda_1(\dot{G}-e_1)\leq \lambda_1(\dot{G}-e_1-e_2)\leq\cdots\leq\lambda_1\Big(\dot{G}-\sum_{i=1}^k e_i\Big),\end{equation*}\] where \(\lambda_1\) denotes the largest eigenvalue, and an inequality in above chain is strict unless the signed graph on the left hand side is disconnected with at least two components with the same largest eigenvalue. We also prove the existence of a connected balanced spanning subgraph \(\dot{H}\) such that \[\begin{equation*}\ell\geq \frac{\big(\lambda_1(\dot{H}\big)-\lambda_1(\dot{G})\big)\big(\Delta+\lambda_1(\dot{H})\big)}{2\Delta}\sqrt{\frac{n}{n-1}},\end{equation*}\] where \(\Delta\) is the maximum vertex degree in \(\dot{H}\), along with the equality precisely in the trivial case when \(\dot{G}\) is balanced. We also derive some consequences of the previous results.