Two Schwenk-like formulas about the signless Laplacian matrix of a graph are given, and thus it gives new tools for computing \(Q\)-
characteristic polynomials of graphs directly. As an application, we give the \(Q\)-characteristic polynomial of lollipop graphs and reprove the known result that no two non-isomorphic lollipop graphs are \(Q\)-cospectral by a simple manner.

In this paper, we give a general result which enlarge the class of graphs known to have \(\alpha\)-labeling.

An independent set in a graph \(G\) is a subset \(I\) of the vertices such that no two vertices in \(I\) are adjacent. We say that \(I\) is a maximum independent set in \(G\) if no other independent set is larger than \(I\). In this paper, we study the problem of determining the second and third largest number of maximum independent sets among all trees and forests. Extremal graphs achieving these values are also given.

This paper is motivated by the concept of the signed \(k\)-independence problem and dedicated to the complexity of the problem on graphs. We show that the problem is linear-time solvable for any strongly chordal graph with a strong elimination ordering and polynomial-time solvable for distance-hereditary graphs. For any fixed positive integer \(k \geq 1\), we show that the signed \(k\)-independence problem on chordal graphs and bipartite planar graphs is NP-complete. Furthermore, we show that even when restricted to chordal graphs or bipartite planar graphs, the signed \(k\)-independence problem, parameterized by a positive integer \(k\) and weight \(\kappa\), is not fixed-parameter tractable.

Edge minimal Hamilton laceable bigraphs on \(2m\) vertices have at least \(\left\lfloor \frac{m+3}{6} \right\rfloor\) vertices of degree \(2\). If a bigraph is edge minimal with respect to Hamilton laceability, it is by definition edge critical, meaning the deletion of any edge will cause it to no longer be Hamilton laceable. The converse need not be true. The \(m\)-crossed prisms \([8]\) on \(4m\) vertices are edge critical for \(m \geq 2\) but not edge minimal since they are cubic. A simple modification of \(m\)-crossed prisms forms a family of “sausage” bigraphs on \(4m + 2\) vertices that are also cubic and edge critical. Both these families share the unusual property that they have exponentially many Hamilton paths between every pair of vertices in different parts. Even so, since the bigraphs are edge critical, deleting an arbitrary edge results in at least one pair having none.

In this paper, we investigate some new identities of symmetry for the Carlitz \(q\)-Bernoulli polynomials invariant under \(S_4\), which are derived from \(p\)-adic \(q\)-integrals on \(\mathbb{Z}_p\).

In this paper, we give the definition of acyclic total coloring and acyclic total chromatic number of a graph. It is proved that the acyclic total chromatic number of a planar graph \(G\) with maximum degree \(\Delta(G)\) and girth \(g\) is at most \(\Delta(G)+2\) if \(\Delta \geq 12\), or \(\Delta \geq 6\) and \(g \geq 4\), or \(\Delta = 5\) and \(g \geq 5\), or \(g \geq 6\). Moreover, if \(G\) is a series-parallel graph with \(\Delta \geq 3\) or a planar graph with \(\Delta \geq 3\) and \(g \geq 12\), then the acyclic total chromatic number of \(G\) is \(\Delta(G) + 1\).

Let \(G\) be a graph and \(\pi(G, x)\) its permanental polynomial. A vertex-deleted subgraph of \(G\) is a subgraph \(G – v\) obtained by deleting from \(G\) vertex \(v\) and all edges incident to it. In this paper, we show that the derivative of the permanental polynomial of \(G\) equals the sum of permanental polynomials of all vertex-deleted subgraphs of \(G\). Furthermore, we discuss the permanental polynomial version of Gutman’s problem [Research problem \(134\), Discrete Math. \(88 (1991) 105–106\)], and give a solution.

A semigraph G is edge complete if every pair of edges in G are adjacent. In this paper, we enumerate the non isomorphic semigraphs in one type of edge complete \((p,3)\) semigraphs without isolated vertices.

In this paper, the \(\lambda\)-number of the circular graph \(C(km, m)\) is shown to be at most \(9\) where \(m \geq 3\) and \(k \geq 2\), and the \(\lambda\)-number of the circular graph \(C(km + s, m)\) is shown to be at most \(15\) where \(m \geq 3\), \(k \geq 2\), and \(1 \leq s \leq m-1\). In particular, the \(\lambda\)-numbers of \(C(2m, m)\) and \(C(n, 2)\) are determined, which are at most \(8\). All our results indicate that Griggs and Yeh’s conjecture holds for circular graphs. The conjecture says that for any graph \(G\) with maximum degree \(\Delta \geq 2\), \(\lambda(G) \leq \Delta^2\). Also, we determine \(\lambda\)-numbers of \(C(n, 3)\), \(C(n, 4)\), and \(C(n, 5)\) if \(n \equiv 0 \pmod{7}\).