We evaluate the convolution sums

\(\sum_{l+30m=n} \sigma(l) \sigma(m), \sum_{3l+10m=n} \sigma(l) \sigma(m),\\ \)
\(\sum_{2l+15m=m} \sigma(l) \sigma(m), \sum_{5l+6m=n} \sigma(l) \sigma(m), \\\)
\(\sum_{l+33m=n} \sigma(l) \sigma(m), \sum_{3l+11m=n} \sigma(l) \sigma(m), \\\)
\(\sum_{l+39m=n} \sigma(l) \sigma(m), \sum_{3l+13m=n} \sigma(l) \sigma(m),\\\)

for all \(n \in \mathbb{N}\) using the theory of quasimodular forms, and utilize these convolution sums to determine the number of representations of a positive integer \(n\) by the forms
\[x_1^2 +x_1x_2+ x_2^2 + x_3^2 +x_3x_4+ x_4^2
+ a(x_5^2 + x_5x_6+x_6^2 + x_7^2 + x_7x_8+x_8^2), \]
for \(a = 10, 11, 13\). Quasimodular forms, divisor functions, convolution sums, representation number \(11A25,11F11,11F25,11F20\)

This paper is an orthogonal continuation of the work of Belbachir and Belkhir in sense where we establish, using bijective proofs, recurrence relations and convolution identities between lines of \(r\)-Lah triangle. It is also established a symmetric function form for the \(r\)-Lah numbers.

For positive integers \(r\) and \(k_1, k_2, \ldots, k_r\), the van der Waerden number \(W(k_1, k_2, \ldots, k_r; r)\) is the minimum integer \(N\) such that whenever the set \(\{1, 2, \ldots, N\}\) is partitioned into \(r\) sets \(S_1, S_2, \ldots, S_r\), there exists a \(k_i\)-term arithmetic progression contained in \(S_i\) for some \(i\). This paper establishes an asymptotic lower bound for \(W(k, m; 2)\) for fixed \(m \geq 3\), improving upon the result of T.C. Brown et al. in [Bounds on some van der Waerden numbers.J. Combin. Theory, Ser.A \(115 (2008), 1304-1309]\). Additionally, we propose lower bounds on certain van der Waerden-like functions.

In this paper, we investigate some properties of higher-order Cauchy of the second kind and poly-Cauchy of the second mixed type polynomials with umbral calculus viewpoint. From our investigation, we derive many interesting identities of higher-order Cauchy of the second kind and poly-Cauchy of the second kind mixed type polynomials.

The chromatic sum \(\Sigma(G)\) of a graph \(G\) is the smallest sum of colors among all proper colorings using natural numbers. In this paper, we establish a necessary condition for the existence of graph homomorphisms. Furthermore, we show that \(\Sigma(G) \leq \chi_f(G) |V(G)|\) holds for every graph \(G\).

The concept of signed cycle domination number of graphs, introduced by B. Xu [B. Xu, On signed cycle domination in graphs, Discrete Math. \(309 (2009)1007-1012]\), is extended to digraphs, denoted by \(\gamma’_{sc}(D)\) for a digraph \(D\). We establish bounds on \(\gamma_s(D)\), characterize all digraphs \(D\) with \(\gamma’_{sc}(D) = |A(D)|-2\), and determine the exact value of \(\gamma’_{sc}(D)\) for specific classes of digraphs \(D\). Furthermore, we define the parameter \(g'(m,n) = \min\{\gamma’_{sc}(D) \mid D \text{ is a digraph with } |V(D)| = n \text{ and } |A(D)| = m\}\) and obtain its value for all integers \(n\) and \(m\) satisfying \(0 \leq m \leq n(n-1)\).

A connected graph \(G\) is \({k-cycle \; resonant}\) if, for \(0 \leq t \leq k\), any \(t\) disjoint cycles \(C_1, C_2, \ldots, C_t\) in \(G\) imply a perfect matching in \(G – \bigcup_{i=1}^{t} V(C_i)\). \(G\) is \({cycle \; resonant}\) if it is \(k^*\)-cycle resonant, where \(k^*\) is the maximum number of disjoint cycles in \(G\). This paper proves that for outerplane graphs, \(2\)-cycle resonant is equivalent to cycle resonant and establishes a necessary and sufficient condition for an outerplanar graph to be cycle resonant. We also discuss the structure of \(2\)-connected cycle resonant outerplane graphs. Let \(\beta(G)\) denote the number of perfect matchings in \(G\). For any \(2\)-connected cycle resonant outerplane graph \(G\) with \(k\) chords, we show \(k+2 \leq \Phi(G) \leq 2^k + 1\) and provide extremal graphs for these inequalities.

In this paper we introduce a new kind of distance Pell numbers which are generated using the classical Fibonacci and Lucas numbers. Generalized companion Pell numbers is closely related to distance Pell numbers which were introduced in \([12]\). We present some relations between distance Pell numbers, distance companion Pell numbers and their connections with the Fibonacci numbers. To study properties of these numbers we describe their graph interpretations which in the special case gives a distance generalization of the Jacobsthal numbers. We also use the concept of a lexicographic product of graphs to obtain a new interpretation of distance Jacobsthal numbers.

For a bipartite graph the extremal number for the existence of a specific odd (even) length path was determined in J. Graph Theory \(8 (1984), 83-95\). In this article, we conjecture that for a balanced bi-partite graph with partite sets of odd order the extremal number for an even order path guarantees many more paths of differing lengths.The conjecture is proved for a linear portion of the conjectured paths.

Let \(D\) be a primitive digraph. Then there exists a nonnegative integer \(k\) such that there are walks of length \(k\) and \(k+1\) from \(u$ to \(v\) for some \(u,v \in V(D)\) (possibly \(u\) again ). Such smallest \(k\) is called the Lewin index of the digraph \(D\), denoted by \(l(D)\). In this paper, the extremal primitive digraphs with both Lewin index \(n — 2\) and girth \(2\) or \(3\) are determined.