Let \(G = (V, E)\) be a graph and \(\phi: V \cup E \to \{1, 2, \ldots, \alpha\}\) be a proper \(\alpha\)-total coloring of \(G\). Let \(f(v)\) denote the sum of the color on vertex \(v\) and the colors on the edges incident with \(v\). A neighbor sum distinguishing \(\alpha\)-total coloring of \(G\) is a proper \(\alpha\)-total coloring of \(G\) such that for each edge \(uv \in E(G)\), \(f(u) \neq f(v)\). Pileeniak and Woźniak first introduced this coloring and conjectured that such coloring exists for any simple graph \(G\) with maximum degree \(\Delta(G)\) if \(\alpha \geq \Delta(G) + 3\). The maximum average degree of \(G\) is the maximum of the average degree of its non-empty subgraphs, which is denoted by \(mad(G)\). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that this conjecture holds for graphs with larger maximum average degree in their list versions. More precisely, we prove that if \(G\) is a graph with \(\Delta(G) \geq 11\) and \(mad(G) < 5\), then \(ch''_{\Sigma}(G) \leq \Delta(G) + 3\), where \(ch''_{\Sigma}(G)\) is the neighbor sum distinguishing total choosability of \(G\).

Let \(\mathcal{K}\) be a family of sets in \(\mathbb{R}^d\) and let \(k\) be a fixed natural number. Assume that every countable subfamily of \(K\) has an intersection expressible as a union of \(k\) starshaped sets, each having a \(d\)-dimensional kernel. Then \(S = \cap \{K : K \in \mathcal{K}\}\) is nonempty and is expressible as a union of \(k\) such starshaped sets.

If members of \(K\) are compact and every finite subfamily of \(\mathcal{K}\) has as its intersection a union of \(k\) starshaped sets, then \(S\) again is a union of \(k\) starshaped sets. An analogous result holds for unions of \(k\) convex sets. Finally, dual results hold for unions of subfamilies of \(\mathcal{K}\).

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.