We give relationships among the binomial coefficients, the Bemoulli numbers and the Stirling numbers, These relations are derived from the translation formulae in the linear discrete systems in Shin-Naito \([8]\).

In this paper, we give the continued fraction expansions of the ordinary generating functions of the derangement polynomials of types \(A\) and \(B\) in a unified manner. Our proof is based on their exponential generating functions and the theory of exponential Riordan arrays.

A graph is called End-regular if its endomorphism monoid is regular. Which graphs are End-regular? This is an open question and difficult to obtain a general answer. In the present paper, we investigate the End-regularity of graphs obtained by adding or deleting vertices from End-regular graphs. As an application, we show that the non-commuting graphs of \(AC\)-groups are End-regular.

In this paper, we study some identities of Barnes-type Genocchi polynomials. We derive those identities by using the fermionic \(p\)-adic integral on \(\mathbb{Z}_p\).
In \([13]\), D.S. Kim and T. Kim established some identities of higher-order Bernoulli and Euler polynomials arising from Bernoulli and Euler basis, respectively. Using the idea developed in \([13]\), we study various identities of special polynomials arising from Barnes-type Genocchi basis.

Suppose that the vertex set of a graph \(G\) is \(V(G) = \{v_1, \ldots, v_n\}\). Then we denote by \({Tr_G}(v_i)\) the sum of distances between \(v_i\) and all other vertices of \(G\). Let \({Tr}(G)\) be the \(n \times n\) diagonal matrix with its \((i,i)\)-entry equal to \({Tr_G}(v_i)\) and \(D(G)\) be the distance matrix of \(G\). Then \(L_p(G) = {Tr}(G) – D(G)\) is the distance Laplacian matrix of \(G\). The largest eigenvalues of \(D(G)\) and \(L_p(G)\) are called distance spectral and distance Laplacian spectral radius of \(G\), respectively. In this paper, we describe the unique graph with maximum distance and distance Laplacian spectral radius among all connected graphs of order \(n\) with given cut edges.

A radio labeling of a connected graph \(G\) of diameter \(d\) is a mapping \(f: V(G) \to \{0, 1, 2, \ldots\}\) such that \(d(u, v) + |f(u) – f(v)| \geq d + 1\) for each pair of distinct vertices \(u\) and \(v\) of \(G\), where \(d(u, v)\) is the distance between \(u\) and \(v\). The value \(rn(f)\) of a radio labeling \(f\) is the maximum label assigned by \(f\) to a vertex of \(G\). The radio number \(rn(G)\) of \(G\) is the minimum value of \(rn(f)\) taken over all radio labelings \(f\) of \(G\). A caterpillar \(C_{m,t}\) is a special tree that consists of a path \(x_1x_2 \ldots x_m\) (\(m \geq 3\)), with some pendant vertices adjacent to the inner vertices \(x_2, x_3, \ldots, x_{m-1}\). If \(d(x_i) = t\) (the degree of \(x_i\)) for \(i = 2, 3, \ldots, m-1\), then the caterpillar is called standard. In this paper, we determine the exact value of the radio number of \(C_{m,t}\) for all integers \(m \geq 4\) and \(t \geq 2\), and explicitly construct an optimal radio labeling. Our results show that the radio number and the construction of optimal radio labeling of paths are special cases of \(C_{m,t}\) with \(t = 2\).

Graph theory, with its diverse applications in theoretical computer science and in natural sciences (chemistry, biology), is becoming an important component of mathematics. Recently, the concepts of new Zagreb eccentricity indices were introduced. These indices were defined for any graph \(G\), as follows: \(M_1^*(G) = \sum_{e_{uv} \in E(G)} [\varepsilon_G(u) + \varepsilon_G(v)]\), \(M_1^{**}(G) = \sum_{v \in V(G)} [\varepsilon_G(v)]^2\), and \(M_2^*(G) = \sum_{e_{uv} \in E(G)} |\varepsilon_G(u) – \varepsilon_G(v)|\), where \(\varepsilon_G(u)\) is the eccentricity value of vertex \(u\) in the graph \(G\). In this paper, new Zagreb eccentricity indices \(M_1^*(G)\), \(M_1^{**}(G)\), and \(M_2^*(G)\) of cycles related graphs, namely gear, friendship, and corona graphs, are determined. Then, a programming code finding values of new Zagreb indices of any graph is offered.

Bizley [J. Inst. Actuar. 80 (1954), 55-62] studied a generalization of Dyck paths from \((0,0)\) to \((pn, gn)\) (\(\gcd(p,q) = 1\)), which never go below the line \(py = qx\) and are made of steps in \(\{(0, 1), (1,0)\}\), called the step set, and calculated the number of such paths. In this paper, we mainly generalize Bizley’s results to an arbitrary step set \(S\). We call these paths \(S\)-\((p,q)\)-Dyck paths, and give explicit enumeration formulas for such paths. In addition, we provide a proof of these formulas using the method presented in Gessel [J. Combin. Theory Ser. A 28 (1980), no. 3, 321-337]. As applications, we calculate some examples which generalize the classical Schröder and Motzkin numbers.

A \(2\)-rainbow dominating function of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all subsets of the set \(\{1,2\}\) such that for any vertex \(v \in V(G)\) with \(f(v) = \emptyset\) the condition \(\bigcup_{u \in N(v)} f(u) = \{1,2\}\) is fulfilled, where \(N(v)\) is the open neighborhood of \(v\). A rainbow dominating function \(f\) is said to be a rainbow restrained domination function if the induced subgraph of \(G\) by the vertices with label \(\emptyset\) has no isolated vertex. The weight of a rainbow restrained dominating function is the value \(w(f) = \sum_{v \in V(G)} |f(v)|\). The minimum weight of a rainbow restrained dominating function of \(G\) is called the rainbow restrained domination number of \(G\). In this paper, we continue the study of the rainbow restrained domination number. First, we classify all graphs \(G\) of order \(n\) whose rainbow restrained domination number is \(n-1\). Then, we establish an upper bound on the rainbow restrained domination number of trees.

The entire chromatic number \(\chi_c(G)\) of a plane graph \(G(V, E, F)\) is the minimum number of colors such that any two distinct adjacent or incident elements receive different colors in \(V(G) \cup E(G) \cup F(G)\). A plane graph \(G\) is called a \(1\)-tree if there exists a vertex \(u \in V(G)\) such that \(G – u\) is a forest. In this paper, it is proved that if \(G\) is a \(2\)-connected \(1\)-tree with \(\Delta(G) \geq 6\), then the entire chromatic number of \(G\) is \(\Delta(G) + 1\), where \(\Delta(G)\) is the maximum degree of \(G\).