Oral Category 5 Abstracts

Disaster preparedness is a cornerstone of disaster risk reduction, especially in the Philippines—one of the most hazard-prone countries in the world. This study aims to develop a statistically sound, location-based Disaster Preparedness Index (DPI) that can be institutionalized as part of the official statistics under the Philippine Statistics Authority (PSA). To support this goal, a statistical operational framework is proposed, ensuring methodological rigor, replicability, and policy relevance. Using a Delphi survey, expert-validated indicators were identified across four domains: Individual and Household Characteristics, Environmental Factors, Governance, and Community Structure and Support Systems. Two approaches—Analytic Hierarchy Process (AHP) and Factor Analysis for Mixed Data (FAMD)—were applied to generate composite DPI scores. A simulation study assessed the statistical properties of both methods.

Results demonstrate that FAMD yields lower bias, standard error, and mean square error across multiple resampling scenarios, indicating superior performance. Governance-related indicators—such as disaster risk reduction plans, local DRRM budget allocations, and institutional mechanisms—emerged as the most influential drivers of preparedness. DPI-based classification of cities and municipalities uncovered significant disparities, with highly urbanized and independent component cities exhibiting stronger disaster readiness than municipalities. This research contributes a data-driven, evidence-based framework for disaster preparedness assessment that can inform national planning, local government strategies, and resource allocation. By aligning with PSA’s standards for official statistics, the proposed DPI and its operational framework can serve as a foundational tool for enhancing the country’s resilience to disasters.

Let G = (V(G), E(G)) be a graph. A function f that assigns to each vertex of G a subset of colors from the set     {1, 2, ..., k}, that is, f : V(G) → P({1, 2, ..., k}), is called a hop k-rainbow dominating function (HkRDF) of G if for every vertex v ∈ V(G) with f(v) = ∅, we have {u ∈ N²_G(v)} f(u) ={1, 2, ..., k}, where N²G(v) is the set of vertices of G at distance two from v. The weight of f, denoted ω(f), is defined as ω(f) = ∑x ∈ V(G) |f(x)|. The hop k- rainbow domination number of G, denoted γhrk(G), is the minimum weight of a hop k-rainbow dominating function of G. A hop k-rainbow dominating function of G with weight γhrk(G) is called a γhrk-function of G. In this paper, we initiate the study of hop k-rainbow domination in graphs. We begin by exploring fundamental properties of this parameter and then establish various bounds on γhrk(G). Furthermore, we identify the graphs for which γhrk(G) = n, and determine exact values for certain graph classes, including complete graphs, complete bipartite graphs, paths, and cycles. Additionally, for any positive integer a, we construct connected graphs satisfying γhr2(G) = γr2(G) = a. Finally, we provide a characterization of all graphs where γhr2(G) = n.

Let G = (V(G), E(G)) be a graph. A function f that assigns to each vertex of G a subset of colors from the set     {1, 2, ..., k}, that is, f : V(G) → P({1, 2, ..., k}), is called a hop k-rainbow dominating function (HkRDF) of G if for every vertex v ∈ V(G) with f(v) = ∅, we have {u ∈ N²_G(v)} f(u) ={1, 2, ..., k}, where N²G(v) is the set of vertices of G at distance two from v. The weight of f, denoted ω(f), is defined as ω(f) = ∑x ∈ V(G) |f(x)|. The hop k- rainbow domination number of G, denoted γhrk(G), is the minimum weight of a hop k-rainbow dominating function of G. A hop k-rainbow dominating function of G with weight γhrk(G) is called a γhrk-function of G. In this paper, we initiate the study of hop k-rainbow domination in graphs. We begin by exploring fundamental properties of this parameter and then establish various bounds on γhrk(G). Furthermore, we identify the graphs for which γhrk(G) = n, and determine exact values for certain graph classes, including complete graphs, complete bipartite graphs, paths, and cycles. Additionally, for any positive integer a, we construct connected graphs satisfying γhr2(G) = γr2(G) = a. Finally, we provide a characterization of all graphs where γhr2(G) = n.

Let G be a matrix group over a field F and ϕ : Mn(F) → Mn(F) be a map such that ϕ(A) ∈ G for all A ∈ G. An element A ∈ G is said to be ϕ−reversible if there exists P ∈ G such that PAP−1 = ϕ(A). If P can be chosen to be an involution (i.e., P2 = I), then A is said to be strongly ϕ−reversible. In this work, we consider the map ϕ : A → −A, and classify the ϕ−reversible and strongly ϕ−reversible elements of the symplectic group.

Let G be a matrix group over a field F and ϕ : Mn(F) → Mn(F) be a map such that ϕ(A) ∈ G for all A ∈ G. An element A ∈ G is said to be ϕ−reversible if there exists P ∈ G such that PAP−1 = ϕ(A). If P can be chosen to be an involution (i.e., P2 = I), then A is said to be strongly ϕ−reversible. In this work, we consider the map ϕ : A → −A, and classify the ϕ−reversible and strongly ϕ−reversible elements of the symplectic group.

Let  be a connected graph on  vertices,  be an -partition, i.e., a sequence of integers forming a partition of  (that is, ). A realization  of  in  is a partition of  into  parts  such that  is a connected graph of order  for every . A graph  is arbitrarily partitionable (AP) if every -partition is realizable in . In this paper, we introduce a variation of this concept which we call almost AP. We define a graph  to be almost AP if there exists exactly one -partition that is not realizable in . In this paper, we give a sufficient condition for a graph to be almost AP and prove that some special graphs are almost AP.

For a simple graph , a double Italian dominating function is a function    having the property  that  for every   ,    if  , then . The weight of a double Italian dominating function is the sum  and the minimum weight of a double Italian dominating function  is the double Italian domination number, denoted by . In this paper, we explore further the concept of double Italian domination. We characterize graphs  with smaller values for  and investigate the double Italian domination on the helm and tadpole graph. Also, we obtain the relationship of the double Italian domination number with the double Roman domination number. Moreover, we investigate the double Italian dominating function of the join, corona, edge corona, and complementary prism of graphs. 

The class of BCK-algebras was introduced by Imai and Iseki in 1966 as a generalization of the concept of set-theoretic difference and propositional calculi. In 2006, Kim introduced the class of KS-semigroups which is both a BCK-algebra and a semigroup. The quotient KS-semigroup via ideals was established and the isomorphism theorems were proved by Cawi and Vilela in 2009.

The notion of fuzzy sets was introduced by Zadeh in 1965 to provide a mathematical framework for dealing with vagueness and imprecision. It departs from classical set theory by allowing elements to have degrees of membership in a set, rather than requiring absolute membership or non-membership.  In 2007, Prince Williams and Husain applied the concept of fuzzy sets to KS-semigroups, and referred to it as a fuzzy KS-semigroup. 

Hyperstructure theory (also called multi-valued algebras) was introduced by Marty at the 8th congress of Scandinavian Mathematicians in 1934. Recall that in a classical algebraic structure, the composition of two elements of a set is an element of the set, while in an algebraic hyperstructure, the composition of two elements is a set. This theory is considered as a generalization of classical algebraic structures. In 2014, Vicedo et al. introduced a new class of algebra related to hyper BCK-algebra and semihypergroup, called hyper KS-semigroup. A quotient hyper KS-semigroup via reflexive hyper KS-ideal was constructed and isomorphism theorems for hyper KS-semigroups were proved.

In this paper, we investigate the concept of fuzzy regular congruence relations on hyper KS-semigroups. We construct a quotient hyper KS-semigroup via fuzzy regular congruence relation and investigate some properties. Moreover, we prove isomorphism theorems on this quotient structure.

The class of BCK-algebras was introduced by Imai and Iseki in 1966 as a generalization of the concept of set-theoretic difference and propositional calculi. In 2006, Kim introduced the class of KS-semigroups which is both a BCK-algebra and a semigroup. The quotient KS-semigroup via ideals was established and the isomorphism theorems were proved by Cawi and Vilela in 2009.

The notion of fuzzy sets was introduced by Zadeh in 1965 to provide a mathematical framework for dealing with vagueness and imprecision. It departs from classical set theory by allowing elements to have degrees of membership in a set, rather than requiring absolute membership or non-membership.  In 2007, Prince Williams and Husain applied the concept of fuzzy sets to KS-semigroups, and referred to it as a fuzzy KS-semigroup. 

Hyperstructure theory (also called multi-valued algebras) was introduced by Marty at the 8th congress of Scandinavian Mathematicians in 1934. Recall that in a classical algebraic structure, the composition of two elements of a set is an element of the set, while in an algebraic hyperstructure, the composition of two elements is a set. This theory is considered as a generalization of classical algebraic structures. In 2014, Vicedo et al. introduced a new class of algebra related to hyper BCK-algebra and semihypergroup, called hyper KS-semigroup. A quotient hyper KS-semigroup via reflexive hyper KS-ideal was constructed and isomorphism theorems for hyper KS-semigroups were proved.

In this paper, we investigate the concept of fuzzy regular congruence relations on hyper KS-semigroups. We construct a quotient hyper KS-semigroup via fuzzy regular congruence relation and investigate some properties. Moreover, we prove isomorphism theorems on this quotient structure.

The Collatz conjecture is a famous problem which lies at the intersection of dynamical systems and number theory. It is focused on the behaviour of the Collatz map T:

Computer simulations show that for any  (approximately ) [1] iterations of Collatz map T starting for n arrive to 1. The conjecture is that this holds true for any natural number. It was initiated (see [2,3]) the study of an operator  acting on Hol(D), the space of all holomorphic functions on the unit disk , defined by . It was shown that the cycles and (hypothetical) diverging trajectories of Collatz map T correspond to classes of fixed points of the operator . In this presentation, we will investigate and analyze structures of fixed points of  on the Hardy space on the unit disk D, . 

Bibliography

[1]   Barina, D. Improved verification limit for the convergence of the Collatz conjecture. J Supercomput 81, 810 (2025). https://doi.org/10.1007/s11227-025-07337-0

[2]    Vincent Béhani. Linear dynamics of an operator associated to the Collatz map. Proceedings of the American Mathematical Society, 2024, 10.1090/proc/16627. hal-04453935

[3] Neklyudov, M. Functional Analysis Approach to the Collatz Conjecture. Results Math 79, 140 (2024). https://doi.org/10.1007/s00025-024-02167-7

An odd-sum coloring of a simple graph  is a proper coloring  such that for each vertex the sum  of all colors used in the closed neighborhood  is odd. The minimum number of colors needed for an odd-sum coloring of a graph G is called the odd-sum chromatic number of G, denoted by  In 2023, Caro et al. posed some interesting problems on odd-sum chromatic number of graphs. One of them states that for every even integer , find graphs G with   and those with . This paper presents some graphs that satisfy the given condition.

Let  be a simple undirected graph. A set  is a geodetic hop dominating set in  if for every , there exist vertices such that  and  lies in a - geodesic, that is,  The minimum cardinality of a geodetic hop dominating set of , denoted by , is called the geodetic hop domination number of . The minimality of  implies that if  such that , then there is at least one vertex of  that is not geodetically hop-dominated by . The -geodetic hop domination defect of , denoted by , is the minimum number of vertices of  that is not geodetically hop-dominated by any subset of vertices of  with cardinality . A set  of cardinality  for which , where , is called a -set of . In this paper, we initiate the study of the concept of -geodetic hop domination defect of a non-trivial graph  and investigate it for some known classes of graphs.

Exchange rate volatility is critical in macroeconomic analysis, influencing financial markets, trade, and policy decisions. This study examines the volatility of the PHP-USD exchange rate from 2008 to 2024, given its economic significance primarily to the Philippines. Investors and traders use volatility as a gauge of risk. Understanding volatility helps investors decide whether they are comfortable with the risk associated with a particular investment. We compare various Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models, including symmetric and asymmetric variants, which capture key characteristics of exchange rate returns. Additionally, we employ the GARCH-Mixed Data Sampling(MiDaS) model to separate short-term and long-term volatility components. To explore macroeconomic influences, we incorporate inflation and net exports as potential drivers of exchange rate volatility. Finally, we incorporate a Long Short-Term Memory (LSTM) approach to the GARCH-MiDaS model to capture both short-term and long-term volatility dynamics. This GARCH-MiDaS-LSTM hybrid model integrates the predictive capabilities of deep neural networks with the structural advantages of mixed-frequency volatility modeling.

Model performance is evaluated using the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Squared Error (MSE), and Mean Absolute Error (MAE). Our findings offer insights into the effectiveness of these models for forecasting exchange rate fluctuations, with implications for financial risk management and economic policy. 

This study integrates the Kalman Filter Smoother-Adaptive Markov Chain Monte Carlo (MCMC) method to estimate the effect of intervention in time series data. Interrupted Time Series (ITS) designs face challenges related to confounding factors and assumptions of linearity. Bayesian Structural Time Series (BSTS) models enhance traditional ITS methods by incorporating predictor variables and leveraging Bayesian inference to estimate counterfactuals. Prior research has employed Gibbs sampling for estimation, but this study introduces an alternative approach by integrating the Kalman Filter and Smoother with Adaptive MCMC.

This adaptive MCMC algorithm, which combines Random Walk Metropolis and Independent Kernel Metropolis-Hastings methods, enhances both convergence and efficiency rates. By refining estimation techniques, this study contributes to the improvement of interrupted time series analysis, ensuring robust and reliable results. The MCMC diagnostics and sensitivity analysis show that the approach is efficient and reliable, with a 97% coverage rate for detecting the true effect, indicating accurate estimation of the intervention effect. The proposed methodology has broad applications in policy evaluation, healthcare intervention, and economics, where accurate assessment of causal impact is crucial for informed decision-making. 

Missing data pose a significant challenge in observational studies particularly in the estimation of treatment effects using propensity score (PS) methods. This study investigates the performance of Predictive Mean Matching (PMM) as a multiple imputation technique for preserving the validity of PS models estimated using logistic regression (LR), gradient boosting machines (GBM), and Bayesian additive regression trees (BART) under conditions of missingness.

Both simulated and empirical datasets were utilized, incorporating two missing data mechanisms: Missing Completely at Random (MCAR) and Missing at Random (MAR). To evaluate the robustness of PMM, varying levels of missingness (up to 15%) were introduced. The analysis focused on assessing the extent to which PMM preserves the accuracy and reliability of PS matching results across different scenarios.

Findings demonstrate that PMM performs well under both MCAR and MAR, with no substantial decrease in model performance across increasing levels of missingness. These results contribute to the growing body of evidence supporting PMM as a practical and robust imputation method. In general, the study shows the importance of appropriate missing data handling to ensure valid and reliable estimation of treatment effects in observational studies.

A toxicokinetic-toxicodynamic (TK-TD) model is developed to assess the impact of mercury on nematode populations, utilizing these organisms as bioindicators for ecological rehabilitation in mercury-contaminated mining areas. Parameter estimation was performed using Approximate Bayesian Computation (ABC) with a trust-region reflective algorithm to fine-tune the model parameters based on experimental data. The incorporation of Michaelis-Menten kinetics refined the model, offering a concentration-responsive depiction of the relationship between mercury exposure and nematode survival. Simulations revealed a significant decline in nematode abundance with increasing mercury concentrations, highlighting its toxic effects and destabilizing influence on populations. The accuracy of the model was further validated through numerical simulations that aligned well with analytical predictions across various scenarios.

This study develops a Bayesian dynamic spatio-temporal model for probabilistic short-term forecasting of solar radiation fields, addressing the need for uncertainty quantification in renewable energy management. The latent solar irradiance field is modeled using a stochastic advection–diffusion partial differential equation (SPDE), integrated within a hierarchical Bayesian framework linking latent states and observational data. Inference combines fixed-lag Ensemble Kalman smoothing and adaptive Markov Chain Monte Carlo (MCMC) methods, enabling efficient sampling in high-dimensional settings. Application to Himawari-9 short-wave radiation (SWR) data over Mindanao, Philippines, demonstrates that normalizing irradiance using the clear-sky index (CSI) improves predictive skill and stability. Posterior estimates indicate strong temporal persistence and moderate diffusion effects, achieving acceptable predictive accuracy for forecast horizons up to 30 minutes. This research represents the first Bayesian formulation of a physically motivated dynamic spatio-temporal model based on advection–diffusion SPDE for high-dimensional solar radiation fields forecasting. The results directly support the United Nations Sustainable Development Goals (SDGs), particularly SDG 7 (Affordable and Clean Energy) and SDG 13 (Climate Action).

Forecasting count time series data in public health surveillance often faces challenges such as zero inflation, overdispersion, and temporal dependence, which traditional models fail to capture. This study addresses these issues by proposing Bayesian Autoregressive Hurdle-INGARCHX(1,1) models. The Poisson and generalized Poisson hurdle models are extended by integrating INGARCHX to include exogenous variables with delay parameters for external influences.

Model parameters are estimated using adaptive Bayesian Markov Chain Monte Carlo (MCMC) methods for efficient and faster convergence. The models are applied to weekly dengue data (2008–2019) from Bislig, Butuan, and Bayugan Cities in the Philippines. Simulation and empirical results show that the Bayesian MCMC approach is effective, with generalized Poisson hurdle-INGARCHX models outperforming the Poisson version based on Deviance Information Criterion (DIC) and Bayes factors. The models yield reliable in-sample predictions, no residual autocorrelation, and low RMSE and MAE in out-of-sample forecasts, despite slight overfitting.

The proposed models support disease forecasting in resource-limited settings and contribute to Sustainable Development Goal 3 (Good Health and Well-Being) by enabling data-driven strategies to reduce dengue and other communicable diseases.

This study introduces the Negative Binomial Hurdle-INGARCHX (NBH-INGARCHX) model to address key challenges in count time series data: overdispersion, excess zeros, and temporal dependence. It combines a hurdle structure to distinguish structural from sampling zeros, an INGARCHX component for autoregressive behavior, and exogenous covariates (rainfall and temperature) for environmental effects. Bayesian estimation is performed using a two-phase Adaptive Markov Chain Monte Carlo (MCMC) algorithm, with convergence validated through trace plots, ACFs, Geweke tests, and inefficiency factors. Simulation and real dengue data from Tandag and Iligan show that NBH-INGARCHX outperforms NB-INGARCHX and ZINB-INGARCHX in handling zero-inflation and improving model fit. Beyond statistical contributions, this work advances SDG 3 (Good Health and Well-Being) by deepening understanding of dengue dynamics, SDG 11 (Sustainable Cities and Communities) through local health planning, and SDG 13 (Climate Action) by integrating climatic drivers. The model offers a practical framework for data-informed public health decisions.

Figure 1. A House Graph G with fd-1G=1.

This study proposes a Bayesian spatio-temporal framework for modeling count data with overdispersion, spatial dependence, and temporal autocorrelation. The NB-INGARCHX model extends Poisson-based approaches by incorporating overdispersion and lagged exogenous covariates, and is structured hierarchically with priors on parameters governing count generation and spatial dynamics. Parameter estimation is performed via an adaptive MCMC algorithm that dynamically tunes proposal distributions to improve convergence and efficiency. Simulation results confirm the model’s robustness, showing accurate parameter recovery across varied scenarios. Model selection metrics further support that model’s performance: SPNB-INGARCH model yielded the lowest DIC value of 25841.51, while SP-NBINGARCHX yielded the highest coverage probability. Diagnostics based on residuals and autocorrelation confirm model adequacy. This work aligns with the United Nations Sustainable Development Goals (SDGs), particularly SDG 3: Good Health and Well-being (Targets 3.3 and 3.D), by enhancing tools for disease surveillance and forecasting. The spatially localized framework also supports SDG 11: Sustainable Cities and Communities, offering guidance for public health planning and targeted interventions in urban and disadvantaged areas.

Contact tracing is an important measure for epidemic control, but information delays can make it less efficient. This study describes a stochastic block model for simulating contact tracing networks with population density and mobility as key parameters. We analyze the graph-theoretic properties of this model, such as degree distribution and clustering coefficient, and show that the simulations are captured by the established theoretical values.

We examine probable COVID-19 contact tracing networks for Hokkaido, Japan, focusing on three scenarios: exclusive consideration of population density, sole focus on mobility, and a combined approach that considers both parameters. Our findings indicate that the combined model generates the most significant results. This is consistent with recent observations in COVID-19 contact tracing networks. 

In 2023, M.-T. Chien, S. Kirkland, C.-K. Li, and H. Nakazato conjectured that the cyclic shift S(a1, a3, a5, ..., a6, a4, a2) with 0 ≤ a1 ≤ ⋯≤ an has the largest numerical range among all cyclic shift matrices with the same collection of weights. H.-L. Gau proved the conjecture in 2024. In this study, a version of the problem for the k-numerical range is formulated. Aspects of the conjecture are explored in this new setting. For small n (at least for n = 2, 3, 4), a necessary and sufficient condition is established for an n-by-n cyclic shift to have the largest k-numerical range among all n-by-n cyclic shift matrices with the same collection of nonnegative real weights. The conjecture is proved for n-by-n cyclic shift matrices. The main results are extended to n-by-n cyclic shift matrices with complex weights. To prove the main results, properties of the k-numerical range are revisited. In particular, it is shown that the k-numerical range is the intersection of certain closed half-planes.

In 2023, M.-T. Chien, S. Kirkland, C.-K. Li, and H. Nakazato conjectured that the cyclic shift S(a1, a3, a5, ..., a6, a4, a2) with 0 ≤ a1 ≤ ⋯≤ an has the largest numerical range among all cyclic shift matrices with the same collection of weights. H.-L. Gau proved the conjecture in 2024. In this study, a version of the problem for the k-numerical range is formulated. Aspects of the conjecture are explored in this new setting. For small n (at least for n = 2, 3, 4), a necessary and sufficient condition is established for an n-by-n cyclic shift to have the largest k-numerical range among all n-by-n cyclic shift matrices with the same collection of nonnegative real weights. The conjecture is proved for n-by-n cyclic shift matrices. The main results are extended to n-by-n cyclic shift matrices with complex weights. To prove the main results, properties of the k-numerical range are revisited. In particular, it is shown that the k-numerical range is the intersection of certain closed half-planes.

In this paper, we introduced and characterized a new class of open set called θβ-open set. Notably, the collection of all θβ-open sets forms a topology. We then examined the relationship between θβ-open sets and other well-known concepts such as classical open sets, θ-open sets, and β-open sets. Additionally, we defined and investigated the concepts of θβ-interior and θβ-closure of a set, as well as θβ-open functions, θβ-closed functions, θβ-continuous functions, and θβ-connectedness. Finally, we present characterizations of θβ-continuous functions from an arbitrary topological space into the product space, along with some versions of separation axioms.

Let G be a connected graph of order n. A set of vertices is considered convex if, for any two vertices withtin the set, all shortest paths between them remain entirely within the set. This study introduces the notion of convex independent neighborhood polynomial of a graph. The convex independent neighborhood polynomial of a graph G of order n in x and y indeterminates, is given by 

where ;  are i-convex subset of G;  is the neighborhood system of i-convex subsets of G;  is the cardinality of the maximum independent set of the graph induced by the neighborhood system of  for each i;  is the number of i-convex subsets of G with maximum independent set of  cardinality equal to j. Specifically, this study establishes some basic properties of the convex independent neighborhood polynomial of a graph, determines the number of convex subsets of some special graphs such as path, cycle, complete graph, complete bipartite, star graph, and of graphs resulting from some binary operation operations, and obtains the convex independent neighborhood polynomial of these graphs. These polynomial are generated by counting the number of convex subsets of a graph with corresponding maximum independent sets in its neighborhood system.

An important concept that can be used to study hop domination in the join and corona of graphs is pointwise non-domination in a graph. Specifically, the concept of pointwise non-dominating set was previously introduced to be able to characterize the hop dominating sets in the join and corona of graphs. To characterize the different variants of hop dominating sets in these graphs, the need to introduce corresponding variants of pointwise non-dominating sets is in order.

Let G be a simple undirected graph with vertex and edge sets V(G) and E(G), respectively. A set S ⊆ V(G) is a pointwise non-dominating set if for each v ∈ V(G) \ S, there exists w ∈ S such that vw ∉ E(G). A pointwise non-dominating set S is movable pointwise non-dominating if for each x ∈ S, S{x} or [S{x}] ∪ {y} for some y ∈ (V(G)\S)∪N(G)(x), is pointwise non-dominating. The minimum cardinality of a movable pointwise non-dominating set, denoted |pnd|m(G), is called the movable pointwise non-domination number of G.

In this paper, we characterize those graphs which admit a pointwise non-dominating set. We give lower and upper bounds on the pointwise non-domination number of a graph and give necessary and sufficient conditions for these bounds to be attained. Values of the movable pointwise non-domination number of some classes of graphs are also obtained. Finally, we use the concept of movable pointwise non-dominating set in a graph to characterize the 2-step movable hop dominating sets in the join of graphs and, subsequently, determine the value and upper bound on the parameter for the join and corona of graphs, respectively.