Let (𝐺) be the eccentricity matrix of a graph 𝐺, and Spec(𝜀(𝐺)) be the eccentricity spectrum of 𝐺. Let [𝐺 1 , 𝐺 2 , … , 𝐺 𝑘 ] be the 𝐻-join of graphs 𝐺 1 , 𝐺 2 , … , 𝐺 𝑘 and let 𝐻[𝐺] be lexicographic product of 𝐻 and 𝐺. This paper determines the eccentricity matrix of a 𝐻-join of graphs. Using this result, we obtain the following results. (i) We construct a family of 𝜀-cospectral graphs, (ii) We derive Spec(𝜀(𝐻[𝐺])) in terms of Spec(𝜀(𝐻)) if rad(𝐻) ≥ 3, where rad(𝐻) is the radius of 𝐺, (iii) We find Spec(𝜀(𝐾 𝑘 [𝐺 1 , 𝐺 2 , … , 𝐺 𝑘 ])) if Δ(𝐺 𝑖 ) ≤ |𝑉 (𝐺 𝑖 )| − 2 which generalizes some of the results from [Mahato et al. in Discrete Appl Math. (2020) 285:252–260], (iv) We determine Spec(𝜀(𝐻[𝐺 1 , 𝐺 2 , … , 𝐺 𝑘 ])) if rad(𝐻) ≥ 2 and 𝐺 𝑖 is complete whenever 𝑒 𝐻 (𝑖) = 2, which generalizes some of the results from [Mahato et al. in Discrete Appl Math. (2020) 285:252–260; Wang et al. in Discrete Math. (2019) 342(9):2636–2646]. Finally, we find the characteristic polynomial of (𝐾 1,𝑚 [𝐺 0 , 𝐺 1 , … , 𝐺 𝑚 ]) if 𝐺 𝑖 ’s are regular. As an application, we deduce some of the results from [Li et al. in Discrete Appl. Math. (2023) 336:47–55; Mahato et al. in Discrete Appl Math. (2020) 285:252–260; Patel et al. in Discrete Math. (2021) 344:112591 and Wang et al. (2018) Discrete Appl Math. 251:299–309].

Krasnosselsky’s art gallery theorem gives a combinatorial characterization of star-shaped sets in Euclidean spaces, similar to Helly’s characterization of finite families of convex sets with non-empty intersection. We study colorful and quantitative variations of Krasnosselsky’s result. In particular, we are interested in conditions on a set 𝐾 that guarantee there exists a measurably large set 𝐾 ′ such that every point in 𝐾 ′ can see every point in 𝐾. We prove results guaranteeing the existence of 𝐾 ′ with large volume or large diameter

Let 𝑓 ∶ 𝑆 𝑞 ⊔ 𝑆 𝑞 → 𝑆 𝑚 be an (ordered oriented) link (i.e. an embedding). How does (the isotopy class of) the knot 𝑆 𝑞 → 𝑆 𝑚 obtained by embedded connected sum of the components of 𝑓 depend on 𝑓? Define a link 𝜎𝑓 ∶ 𝑆 𝑞 ⊔ 𝑆 𝑞 → 𝑆 𝑚 as follows. The first component of 𝜎𝑓 is the ‘standardly shifted’ first component of 𝑓 . The second component of 𝜎𝑓 is the embedded connected sum of the components of 𝑓 . How does (the isotopy class of) 𝜎𝑓 depend on 𝑓 ? How does (the isotopy class of) the link 𝑆 𝑞 ⊔ 𝑆 𝑞 → 𝑆 𝑚 obtained by embedded connected sum of the last two components of a link 𝑔 ∶ 𝑆 𝑞 1 ⊔ 𝑆 𝑞 2 ⊔ 𝑆 𝑞 3 → 𝑆 𝑚 depend on 𝑔? We give the answers for the ‘first non-trivial case’ 𝑞 = 4𝑘−1 and 𝑚 = 6𝑘. The first answer was used by S. Avvakumov for classification of linked 3-manifolds in 𝑆 6 .

We introduce the 𝐵-Stirling numbers of the first and second kinds, which are the coefficients of the potential polynomials when we express them in terms of the monomials and the falling factorials, respectively. These numbers include, as particular cases, the partial and complete Bell polynomials, the degenerate and probabilistic Stirling numbers, and the 𝑆-restricted Stirling numbers, among others. Special attention is devoted to the computation of such numbers. On the one hand, a recursive formula is provided. On the other hand, we can compute Stirling numbers of one kind in terms of the other, with the help of the classical Stirling numbers.

Extending work of Saneblidze–Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of 𝐴 ∞ -algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue of twisted complexes (type DD structures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of 𝐴 ∞ -algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology.

Based on two involutions and a bijection, we combinatorially determine the difference between the number of 𝓁-regular partitions of 𝑛 into an even number of parts and into an odd number of parts for all positive integers 𝑛 and 𝓁 > 1, which extends two recent results due to Ballantine and Merca. As an application, we provide a combinatorial proof of Hickerson’s identity on the number of partitions into an even and odd number of parts.

Let 𝐺 be a connected 𝐾1,5-free graph with 𝑛 vertices. In this paper, we study some optimal sufficient conditions for a connected 𝐾1,5-free graph to have a spanning tree with few leaves and branch vertices in total. In particular, we first prove that if 𝜎5(𝐺) ≥ 𝑛 − 2, then 𝐺 contains a spanning tree with at most seven leaves and branch vertices. After that, we show all graphs 𝐺 which have no spanning tree with at most seven leaves and branch vertices and 𝜎5(𝐺) = 𝑛 − 3.

A question of Erdős asked whether there exists a set of 𝑛 points such that 𝑐 ⋅ 𝑛 distances occur more than 𝑛 times. We provide an affirmative answer to this question, showing that there exists a set of 𝑛 points such that distances occur more than 𝑛 times. We also present a generalized version, finding a set of 𝑛 points where 𝑐𝑚 ⋅ 𝑛 distances occurring more than 𝑛 + 𝑚 times.

A long standing Total Coloring Conjecture (TCC) asserts that every graph is total colorable using its maximum degree plus two colors. A graph is type-1 (or type-2) if it has a total coloring using maximum degree plus one (or maximum degree plus two) colors. For a graph 𝐺 with 𝑚 vertices and for a family of graphs {𝐻1, 𝐻2, … , 𝐻𝑚}, denote , the generalized corona product of 𝐺 and 𝐻1, 𝐻2, … , 𝐻𝑚. In this paper, we prove that the total chromatic number of is the maximum of total chromatic number of 𝐺 and maximum degree of plus one. As an immediate consequence, we prove that is type-1 when 𝐺 satisfies TCC and also the corona product of 𝐺 and 𝐻 is type-1 if 𝐺 satisfies TCC. This generalizes some results in (R. Vignesh. et. al. in Discrete Mathematics, Algorithms and Applications, 11(1): 2019) and all the results in (Mohan et. al. in Australian Journal of Combinatorics, 68(1): 15-22, 2017.)

We revisit the algorithmic problem of finding a triangle in a graph (Triangle Detection), and examine its relation to other problems such as 3Sum, Independent Set, and Graph Coloring. We obtain several new algorithms:(I) A simple randomized algorithm for finding a triangle in a graph. As an application, we study a question of Pˇatraşcu (2010) regarding the triangle detection problem.(II) An algorithm which given a graph 𝐺 = (𝑉 , 𝐸) performs one of the following tasks in 𝑂(𝑚 + 𝑛) (i.e., linear) time: (i) compute a Ω(1/√𝑛)-approximation of a maximum independent set in 𝐺 or (ii) find a triangle in 𝐺. The run-time is faster than that for any previous method for each of these tasks.(III) An algorithm which given a graph 𝐺 = (𝑉 , 𝐸) performs one of the following tasks in 𝑂(𝑚+𝑛3/2) time: (i) compute √𝑛-approximation for Graph Coloring of 𝐺 or (ii) find a triangle in 𝐺. The run-time is faster than that for any previous method for each of these tasks on dense graphs, with 𝑚 = (𝑛9/8).(IV) Results (II) and (III) above suggest the following broader research direction: if it is difficult to find (A) or (B) separately, can one find one of the two efficiently? This motivates the dual pair concept we introduce. We provide several instances of dual-pair approximation, relating Longest Path, (1,2)-TSP, and other NP-hard problems.