Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...

Let μ (G) be the largest eigenvalue of a graph G and Tr (n) be the r-partite Turán graph of order n. We prove that if G is a graph of order n with μ (G) > μ (Tr (n)) , then G contains various large supergraphs of the complete graph of order r + 1, e.g., the complete r-partite graph with all parts of size log n with an edge added to the first part. We also give corresponding stability results.

Let $G$ be a connected graph and let $T$ spanning tree of $G$. A partial orientation $\sigma$ respect to is an the edges except those $T$, resulting associated with which denoted by $G_T^\sigma$. In this paper we prove that there exists such largest eigenvalue Hermitian adjacency matrix $G_T^\sigma$ at most absolute value roots matching polynomial

Algebraic combinatorics is the area of mathematics that uses the theories and methods of abstract and linear algebra to solve combinatorial problems, or conversely applies combinatorial techniques to solve problems in algebra. In particular, spectral graph theory applies the techniques of linear algebra to study graph theory. Spectral graph theory is the study of the eigenvalues of various matr...

for any $k in mathbb{n}$, the $k$-subdivision of graph $g$ is a simple graph $g^{frac{1}{k}}$, which is constructed by replacing each edge of $g$ with a path of length $k$. in [moharram n. iradmusa, on colorings of graph fractional powers, discrete math., (310) 2010, no. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $g$ has been introduced as a fractional power of $g$, denoted by ...

Among all connected cactuses with n vertices we find a unique graph whose largest eigenvalue (index, for short) is maximal.

The problem of partitioning a graph such that the number of edges incident to vertices in diierent partitions is minimized, arises in many contexts. Some examples include its recursive application for minimizing ll-in in matrix factorizations and load-balancing for parallel algorithms. Spectral graph partitioning algorithms partition a graph using the eigenvector associated with the second smal...