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.