Contributions to Discrete Mathematics (E-Journal, University of Calgary)

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On the double covers of a line graph.

Author: Chauhan Shivani
Reddy A.Satyanarayana
Publication venue: Faculty of Science, University of Calgary
Publication date: 30/04/2025
Field of study

Let

L(X)

be the line graph of graph

X

. Let

X^{\prime\prime}

be the Kronecker product of

X

K_2

. In this paper, we see that

L(X^{\prime\prime})

is a double cover of

L(X)

. We define the symmetric edge graph of

X

, denoted as

\rm{ga}(X)

which is also a double cover of

L(X)

. We study various properties of

\rm{ga}(X)

in relation to

X

and the relationship amongst the three double covers of

L(X)

that are

L(X^{\prime\prime}),\rm{ga}(X)

and

L(X)^{\prime\prime}

. With the help of these double covers, we show that for any integer

k\geq 5

, there exist two equienergetic graphs of order

2k

that are not cospectral

Dyck paths with first sojourn highest

Author: Blecher Aubrey
Knopfmacher Arnold
Publication venue: Faculty of Science, University of Calgary
Publication date: 28/10/2025
Field of study

A Dyck path is a lattice path in the first quadrant using steps

u=(1,1)

and

d=(1,-1)

, starting at the origin and ending on the

x

-axis. A Dyck sojourn is a Dyck path with only one return to the

x

-axis. Various subclasses of Dyck paths are shown to be counted by Catalan numbers: Dyck paths with weakly highest first sojourn and semilength

n+1

are counted by twice the

n

th Catalan number and Dyck paths with semilength

n+1

where the second sojourn is weakly highest are counted by the

n

th Catalan number. Also Dyck paths in which the first sojourn is strictly highest are equinumerous to Dyck paths with exactly one peak of maximum height. Some of these equivalences are proved using bijections. Generating functions for all these subclasses are also developed in the paper. The asymptotics as

n\to\infty

for the case where the first sojourn is strictly highest are explored

A generalization of the Beraha–Kahane–Weiss theorem with graph polynomial applications

Author: Brown Jason
Otto Peter
Publication venue: Faculty of Science, University of Calgary
Publication date: 28/10/2025
Field of study

The beautiful Beraha–Kahane–Weiss (BKW) theorem has found many applications within graph theory, allowing for the determination of the limits of zeros of graph polynomials in a wide range of settings such as chromatic polynomials, network reliability, and generating polynomials related to independence and domination. However, the proof only provides solutions for linear recurrence relations of polynomials whose characteristic polynomials have simple zeros. Here we extend the class of functions to which the BKW theorem can be applied, and provide some applications in combinatorics

(G, s)-Transitive Graphs Of Valency 15

Author: Abedi Fateme
Salarian M. Reza
Publication venue: Faculty of Science, University of Calgary
Publication date: 30/04/2025
Field of study

Let

X

be a finite simple undirected graph and

G\leq \rm{Aut}(X)

. If

G

is transitive on the set of

s

-arcs but not on the set of

(s+1)

-arcs of

X

, then

X

is called

(G, s)

-transitive. In this paper, we determine the structure of the vertex-stabilizer

G_{v}

when

X

is a connected

(G,s)

-transitive graph of valency

15

. We also give some examples to show that each type of

G_{v}

with

s\ge 2

, can be realized

On Freiman-Lev conjecture

Author: Wang Yujie
Tang Min
Publication venue: Faculty of Science, University of Calgary
Publication date: 30/04/2025
Field of study

Let

A

be a set of

k

integers such that

A\subseteq [0, l]

0, l\in A

and

\gcd(A)=1

. Let

2^{\wedge}A

denote the set of all sums of two distinct elements of

A

. Write

W=\{w\in [0, l]\backslash A: w,w+l\not\in 2^{\wedge}A\}

. In this paper, we obtain the upper bound of

|W|

with some restrictions on

l

. As an application, we show that the Freiman-Lev conjecture is true for

l=2k-4

using the structure of

A

with

|W|=2

Applications via series accelerations of new identities involving Catalan-type numbers

Author: Campbell John M.
Kilic Emrah
Publication venue: Faculty of Science, University of Calgary
Publication date: 28/10/2025
Field of study

We introduce infinite families of terminating hypergeometric identities involving generalizations of Catalan numbers, generalizing results introduced by Chu and K\i l\i\c{c}, and we apply Wilf—Zeilberger (WZ) pairs associated with our new identities via a series acceleration method. We apply a WZ pair introduced in our article to prove an identity for accelerating the convergence for a family of

{}_{3}F_{2}(1)

-series with three real parameters from

1

\frac{1}{4}

, and we apply this identity to generalize Ramanujan-like series for

\frac{1}{\pi}

\frac{\sqrt{2}}{\pi}

\frac{\sqrt{3}}{\pi}

, and

\frac{\sqrt{2 \pm \sqrt{2}}}{\pi }

that are due to Chu et al. A fast-converging series for

\pi^2

due to Guillera is also a special case of our acceleration identity. We also apply another WZ pair introduced in this article to prove an identity for accelerating the convergence of a

{}_{3}F_{2}(1)

-family with three real parameters from

1

\frac{1}{16}

, and we apply this result, via a series bisection, to formulate a new WZ proof of Ramanujan\u27s series for

\frac{1}{\pi}

of convergence rate

\frac{1}{4}

. A number of our finite sums involving Catalan-type numbers are such that up-to-date versions of the Maple Computer Algebra System cannot compute WZ pairs for such sums, which is representative of the computationally challenging nature of our results

On next-to-minimum size blocking sets of external lines to a nondegenerate quadric in $\operatorname{PG}(3,q)$

Author: De Bruyn Bart
Pradhan Puspendu
Sahoo Binod Kumar
Publication venue: Faculty of Science, University of Calgary
Publication date: 28/10/2025
Field of study

Consider a hyperbolic or an elliptic quadric

Q^\epsilon(3,q)

\epsilon \in \{ +,- \}

, in

\operatorname{PG}(3,q)

and let

\mathcal{E}^\epsilon

denote the set of all lines of

\operatorname{PG}(3,q)

that are external with respect to

Q^\epsilon(3,q)

. If

\pi

is a (tangent or secant) plane of

\operatorname{PG}(3,q)

, then

\pi \setminus Q^\epsilon(3,q)

is an

\mathcal{E}^\epsilon

-blocking set which is minimal, except when

(q,\epsilon) = (2,+)

and

\pi

is a secant plane. In this way, we obtain two families of (minimal) blocking sets of sizes

m_1^\epsilon

and

m_2^\epsilon

, with

(m_1^+,m_2^+)=(q^2-q,q^2)

and

(m_1^-,m_2^-)=(q^2,q^2+q)

, where those of size

m_1^\epsilon

are also the minimum size

\mathcal{E}^\epsilon

-blocking sets. Motivated by the search for new (families of) minimal

\mathcal{E}^\epsilon

-blocking sets with sizes in the open interval

]m_1^\epsilon,m_2^\epsilon[

, we determine here all

\mathcal{E}^\epsilon

-blocking sets of size

m_1^\epsilon+1

\operatorname{PG}(3,q)

in the case

q \in \{ 2,3 \}

Orthogonal colourings of tensor graphs

Author: MacKeigan Kyle
Publication venue: Faculty of Science, University of Calgary
Publication date: 28/10/2025
Field of study

Perfect

k

-orthogonal colourings of tensor product graphs are studied in this article. First, the problem of determining if a given graph has a perfect 2-orthogonal colouring is reformulated as a tensor subgraph problem. Then, it is shown that if two graphs have a perfect

k

-orthogonal colouring, then so does their tensor graph. This provides an upper bound on the

k

-orthogonal chromatic number for general tensor graphs. Lastly, two other conditions for a tensor graph to have a perfect

k

-orthogonal colouring are given

Identities involving trigonometric, zeta and partition functions

Author: Alegri Mateus
Da Silva Robson
Santos Wagner Ferreira
Publication venue: Faculty of Science, University of Calgary
Publication date: 28/10/2025
Field of study

Using well-known trigonometric functions, we establish identities involving the multiple zeta function and the multiple lambda function. Furthermore, we derive new identities by applying classical trigonometric relations. Some of these results are expressed in terms of colored partitions, highlighting connections between partition theory and special functions

Congruence properties of 8- and 9-colored generalized Frobenius partitions modulo 5

Author: Tang Dazhao
Publication venue: Faculty of Science, University of Calgary
Publication date: 28/10/2025
Field of study

In his 1984 AMS Memoir, Andrews introduced the family of functions

c\phi_k(n)

, which denotes the number of

k

-colored generalized Frobenius partitions of

n

. In this paper, by employing some

q

-series identities and elementary generating function manipulations, we prove a characterization of

c\phi_8(5n+2)

modulo

5

. Moreover, we derive a characterization of

c\phi_9(5n+1)

modulo

5

. These two characterizations can lead to the corresponding infinite sets of Ramanujan-type congruences modulo

5

satisfied by

c\phi_8(n)

and

c\phi_9(n)

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Contributions to Discrete Mathematics (E-Journal, University of Calgary)

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