Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. Recently, we introduced a new invariant of a graph \(G\), denoted as \(R_5(G)\). Using this invariant and the properties of the adjoint polynomials, we completely determine the adjoint equivalence class of \(\psi_n^3({n-3,1})\). According to the relations between adjoint polynomial and chromatic polynomial, we also simultaneously determine the chromatic equivalence class of \(\psi_n^3({n-3,1})\).

In this article, we prove a conjecture about the equality of two generating functions described in “From Parking Functions to Gelfand Pairs” (Aker, Can, 2012) attached to two sets whose cardinalities are given by Catalan numbers. We establish a combinatorial bijection between the two sets on which the two generating functions were based.

Let \(G\) be a finite cyclic group. Every sequence \(S\) of length \(l\) over \(G\) can be written in the form \(S = (x_1g) + \cdots + (x_lg)\), where \(g \in G\) and \(x_1, \ldots, x_l \in [1, ord(g)]\), and the index \(ind(S)\) of \(S\) is defined to be the minimum of \((x_1 + \cdots + x_l)/ord(g)\) over all possible \(g \in G\) such that \(\langle g \rangle = G\). Recently, the second and third authors determined the index of any minimal zero-sum sequence \(S\) of length \(5\) over a cyclic group of a prime order where \(S =g^2 \cdot (x_2g)\cdot (x_3g)\cdot (x_4g)\). In this paper, we determine the index of any minimal zero-sum sequence \(S\) of length \(5\) over a cyclic group of a prime power order. It is shown that if \(G = \langle g \rangle\) is a cyclic group of prime power order \(n = p^{\mu}\) with \(p \geq 7\) and \(\mu \geq 2\), and \(S = (x_1g) \cdot (x_2g) \cdot (x_3g) \cdot (x_4g) \cdot (x_5g)\) with \(x_1 = x_2\) is a minimal zero-sum sequence with \(\gcd(n, x_1, x_2, x_3, x_4, x_5) = 1\), then \(ind(S) = 2\) if and only if \(S = (mg) \cdot (mg) \cdot (m\frac{n-1}{2}g) \cdot (m\frac{n+3}{2}g) \cdot (m(n-3)g)\) where \(m\) is a positive integer such that \(\gcd(m,n) = 1\).

Let \(G\) be a graph with vertex set \(V(G)\). For any integer \(k \geq 1\), a signed \(k\)-dominating function is a function \(f: V(G) \rightarrow \{-1, 1\}\) satisfying \(\sum_{x \in N[v]} f(t) \geq k\) for every \(v \in V(G)\), where \(N[v]\) is the closed neighborhood of \(v\). The minimum of the values \(\sum_{v \in V(G)} f(v)\), taken over all signed \(k\)-dominating functions \(f\), is called the signed \(k\)-domination number. In this note, we present some new lower bounds on the signed \(k\)-domination number of a graph. Some of our results improve known bounds.

In this paper, we define and study the \(k\)-order Gaussian Fibonacci and Lucas numbers with boundary conditions. We identify and prove the generating functions, the Binet formulas, the summation formulas, matrix representation of \(k\)-order Gaussian Fibonacci numbers, and some significant relationships between \(k\)-order Gaussian Fibonacci and \(k\)-order Lucas numbers, connecting them with usual \(k\)-order Fibonacci numbers.

A vertex-colored path is vertex-rainbow if its internal vertices have distinct colors. For a connected graph \(G\) with connectivity \(\kappa(G)\) and an integer \(k\) with \(1 \leq k \leq \kappa(G)\), the rainbow vertex \(k\)-connectivity of \(G\) is the minimum number of colors required to color the vertices of \(G\) such that any two vertices of \(G\) are connected by \(k\) internally vertex-disjoint vertex-rainbow paths. In this paper, we determine the rainbow vertex \(k\)-connectivities of all small cubic graphs of order \(8\) or less.