Circulant matrices play an important role in solving ordinary and partial differential equations. In this paper, by using the inverse factorization of polynomial of degreen, the explicit determinants of circulant and left circulant matrix involving Tribonacci numbers or generalized Lucas numbers are expressed in terms of Tribonacci numbers and generalized Lucas numbers only. Furthermore, four kinds of norms and bounds for the spread of these matrices are given, respectively.
A brick is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is $b$-invariant if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is essentially 4-edge-connected if it does not contain nontrivial 3-cuts. A brick $G$ is near-bipartite if it has a pair of edges $\{e_1, e_2\}$ such that $G-\{e_1,e_2\}$ is bipartite and matching covered.
Kothari, de Carvalho, Lucchesi and Little proved that each essentially 4-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges. Moreover, they made a conjecture: every essentially 4-edge-connected cubic near-bipartite brick $G$, distinct from $K_4$, has at least $|V(G)|/2$ $b$-invariant edges. We confirm the conjecture in this paper. Furthermore, all the essentially 4-edge-connected cubic near-bipartite bricks, the numbers of $b$-invariant edges of which attain the lower bound, are presented.
A quadrilateral cylinder of length $m$ and breadth $n$ is the Cartesian product of a $m$-cycle(with $m$ vertices) and a $n$-path(with $n$ vertices). Write the vertices of the two cycles on the boundary of the quadrilateral cylinder as $x_1,x_2,\cdots,x_m$ and $y_1,y_2,\cdots ,y_m$, respectively, where $x_i$ corresponds to $y_i(i=1,2,\dots, m)$. We denote by $Q_{m,n,r}$, the graph obtained from quadrilateral cylinder of length $m$ and breadth $n$ by adding edges $x_iy_{i+r}$ ($r$ is a integer, $0\leq r < m$ and $i+r$ is modulo $m$). Kasteleyn had derived explicit expressions of the number of perfect matchings for $Q_{m,n,0}$ [P.W. Kasteleyn, The statistics of dimers on a lattice I: The number of dimer arrangements on a quadratic lattice, Physica 27(1961), 1209–1225]. In this paper, we generalize the result of Kasteleyn, and obtain expressions of the number of perfect matchings for $Q_{m,n,r}$ by enumerating Pfaffians.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.